Both forms on the right-hand side of Eq. (H) are equivalent [49], but using
the second form with Eq. (146) it follows immediately that

$\mathcal{F}_{\lambda}[P(\bm{\sigma},\mathbf{t}|t,\lambda)]=\frac{{\rm
E}[N|t,\lambda]}{1+\lambda}.$ (150)

###  H.1 Sequence observables

To recover the KUR, consider an observable $\Theta(\bm{\sigma})$, which
depends only on the sequence $\bm{\sigma}$ and not the jump times
$\mathbf{t}$. Its expectation value is ${\rm
E}[\Theta|t,\lambda]=\sum_{\bm{\sigma}}P(\bm{\sigma}|t,\lambda)\Theta(\bm{\sigma})$,
where

$\displaystyle P(\bm{\sigma}|t,\lambda)$ $\displaystyle=\int_{0}^{t}{\rm
d}t_{N}\cdots\int_{0}^{t_{2}}{\rm d}t_{1}P(\bm{\sigma},\mathbf{t}|t,\lambda)$
$\displaystyle=\int_{0}^{(1+\lambda)t}{\rm
d}t^{\prime}_{N}\cdots\int_{0}^{t^{\prime}_{2}}{\rm
d}t^{\prime}_{1}P\left(\bm{\sigma},\mathbf{t}^{\prime}|(1+\lambda)t\right)$
$\displaystyle=P\left(\bm{\sigma}|(1+\lambda)t\right).$ (151)

On the second line we changed integration variables to
$t_{j}^{\prime}=(1+\lambda)t_{j}$, and we omit the explicit conditioning on
$\lambda$ when writing quantities with $\lambda=0$, e.g.
$P(\bm{\sigma}|t)=P(\bm{\sigma}|t,\lambda=0)$. Therefore, for such expectation
values, rescaling the generator $\mathsf{L}\to(1+\lambda)\mathsf{L}$ is
equivalent to rescaling the total time as $t\to(1+\lambda)t$. Explicitly, we
can write

${\rm E}[\Theta|t,\lambda]={\rm E}[\Theta|(1+\lambda)t]={\rm
E}[\Theta|t]+\lambda t\partial_{t}{\rm E}[\Theta|t]+\mathcal{O}(\lambda^{2}).$
(152)

Plugging Eqs. (152) and (150) into the CRB (148), taking the limit $\lambda\to
0$, and rearranging, we get the KUR

$\mathcal{S}=\frac{(\partial_{t}{\rm E}[\Theta|t])^{2}}{{\rm
Var}[\Theta|t]/t}\leq\mathcal{A},$ (153)

where $\mathcal{A}={\rm E}[N|t]/t=\sum_{\mu,\sigma}{\rm
E}[N_{\mu\sigma}|t]/t=\sum_{\mu,\sigma}R_{\mu\sigma}p_{\sigma}^{{\rm ss}}$ is
the dynamical activity.

###  H.2 Waiting-time observables

For observables that depend non-trivially on the waiting times $\bm{\tau}$,
the perturbed expectation value (147) is given by

$\displaystyle{\rm E}\left[\Theta|t,\lambda\right]$
$\displaystyle=\sum_{\bm{\sigma}}\int_{0}^{t}{\rm
d}\mathbf{t}\,P(\bm{\sigma},\mathbf{t}|t,\lambda)\Theta(\bm{\sigma},\bm{\tau})$
$\displaystyle=\sum_{\bm{\sigma}}\int_{0}^{(1+\lambda)t}{\rm
d}\mathbf{t}^{\prime}\,P(\bm{\sigma},\mathbf{t}^{\prime}|(1+\lambda)t)\Theta\left(\bm{\sigma},(1+\lambda)^{-1}\bm{\tau}^{\prime}\right)$
$\displaystyle={\rm
E}\left[\Theta\left(\bm{\sigma},(1+\lambda)^{-1}\bm{\tau}\right)\big{|}(1+\lambda)t\right],$
(154)

where on the second line we changed variables to
$t^{\prime}_{j}=(1+\lambda)t_{j}$, as in Eq. (H.1), and the notation
$(1+\lambda)^{-1}\bm{\tau}$ means that each waiting time is rescaled as
$\tau_{j}\to(1+\lambda)^{-1}\tau_{j}$. Expanding for small $\lambda$ yields

$\partial_{\lambda}{\rm E}\left[\Theta|t,\lambda\right]=t\partial_{t}{\rm
E}[\Theta|t]-{\rm
E}\left[\sum_{j=0}^{N}\tau_{j}\partial_{\tau_{j}}\Theta(\bm{\sigma},\bm{\tau})\Bigg{|}t\right]+\mathcal{O}(\lambda),$
(155)

which reproduces Eq. (36). As a consequence, the arguments leading to the KUR
(153) no longer hold in general.

###  H.3 Observables linear in the waiting times $\tau_{j}$

As an example of the general observables discussed in Appendix H.2 above,
consider an observable that is a linear function of the waiting times,
$\Theta(\bm{\sigma},\bm{\tau})=\sum_{j=0}^{N}a_{j}(\bm{\sigma})\tau_{j}$ with
$\\{a_{j}(\bm{\sigma})\\}$ arbitrary functions of $\bm{\sigma}$. We then have
$\sum_{j=0}^{N}\tau_{j}\partial_{\tau_{j}}\Theta=\Theta$. If the observable is
also time-extensive, ${\rm E}[\Theta|t]\sim t$, then for large times
$t\partial_{t}{\rm E}[\Theta|t]={\rm E}[\Theta|t]$. Under these conditions,
the right-hand side of Eq. (155) vanishes up to $\mathcal{O}(\lambda)$
corrections and thus, for $\lambda\to 0$, the CRB (148) reduces to the trivial
bound ${\rm Var}[\Theta]\geq 0$. As discussed in Sec. III.3, this bound is
saturated by the observable $\Theta(\bm{\tau})=\sum_{j=0}^{N}\tau_{j}=t$.
Since the total time $t$ is fixed, its variance vanishes.

A non-trivial example of such an observable is given by

$\Theta(\bm{\sigma},\bm{\tau})=\sum_{j=0}^{N}\Gamma_{\sigma_{j}}\tau_{j}\equiv
Q(\bm{\sigma},\bm{\tau}).$ (156)

The statistics of this observable can be found using the following convenient
trick. Let us define the moment-generating functions for $Q$ and for the
number of jumps, $N$, as

$\displaystyle M_{Q}(\xi,t)={\rm E}\left[{\rm e}^{-\xi Q}\big{|}t\right],$
(157) $\displaystyle M_{N}(\eta,t)={\rm E}\left[{\rm e}^{-\eta
N}\big{|}t\right],$ (158)

where $\xi$ and $\eta$ are counting parameters. To derive a relation between
these two functions, we write out Eq. (157) explicitly using Eq. (54) and the
notation of Eq. (A), finding

$\displaystyle M_{Q}(\xi,t)$ $\displaystyle=\sum_{\bm{\sigma}}\int_{0}^{t}{\rm
d}\mathbf{t}\,{\rm
e}^{-(1+\xi)Q(\bm{\sigma},\bm{\tau})}\prod_{j=0}^{N-1}R_{\sigma_{j+1}\sigma_{j}}p_{\sigma_{0}}^{\rm
ss}$
$\displaystyle=\sum_{N=0}^{\infty}(1+\xi)^{-N}\\!\sum_{\sigma_{0}=1}^{d}\cdots\sum_{\sigma_{N}=1}^{d}\int_{0}^{(1+\xi)t}\\!\\!{\rm
d}\mathbf{t}^{\prime}P\big{(}\bm{\sigma},\mathbf{t}^{\prime}|(1+\xi)t\big{)}$
$\displaystyle=\sum_{N=0}^{\infty}(1+\xi)^{-N}P\big{(}N|(1+\xi)t\big{)}$
$\displaystyle={\rm E}\left[(1+\xi)^{-N}\big{|}(1+\xi)t\right].$ (159)

The second equality above follows after introducing new integration variables
$t_{j}^{\prime}=(1+\xi)t_{j}$, and in the third equality we recognise the
total probability for $N$ jumps in a time $(1+\xi)t$, defined by Eq. (56).
Comparing Eq. (H.3) with Eq. (158), we deduce the relation

$M_{Q}(\xi,t)=M_{N}\big{(}\ln(1+\xi),(1+\xi)t\big{)}.$ (160)

This shows that the moments of $Q$ can be expressed in terms of moments of
$N$, and vice versa.

Now, an explicit expression for $M_{N}(\eta,t)$ can be found by noting that
${\rm e}^{-\eta N}=\prod_{j=0}^{N-1}{\rm e}^{-\eta}$, and therefore

$\displaystyle M_{N}(\eta,t)$
$\displaystyle=\sum_{\bm{\sigma}}\int_{0}^{t}{\rm d}\mathbf{t}\,{\rm
e}^{-Q(\bm{\sigma},\bm{\tau})}\prod_{j=0}^{N-1}{\rm
e}^{-\eta}R_{\sigma_{j+1}\sigma_{j}}p_{\sigma_{0}}^{\rm ss}$
$\displaystyle=\mathbf{u}^{\sf T}\mathsf{G}_{\eta}(t)\mathbf{p}^{\rm ss},$
(161)

where we resummed the Dyson series (cf. Eq. (52)) to obtain the “tilted”
propagator $\mathsf{G}_{\eta}(t)=\exp\left[\left({\rm
e}^{-\eta}\mathsf{R}-\mathsf{\Gamma}\right)t\right]$. This result is well
known from the theory of full counting statistics [44]. Using
$\mathsf{\Gamma}=\mathsf{R}-\mathsf{L}$ and defining
$\mathsf{\check{R}}(t)={\rm e}^{-\mathsf{L}t}\mathsf{R}{\rm e}^{\mathsf{L}t}$,
we can write the tilted propagator in terms of a time-ordered exponential as

$\mathsf{G}_{\eta}(t)=\mathsf{G}(t)\,\overleftarrow{\mathrm{T}}\exp\left[\int_{0}^{t}{\rm
d}t^{\prime}\left({\rm
e}^{-\eta}-1\right)\mathsf{\check{R}}(t^{\prime})\right].$ (162)

Finally, to obtain $M_{Q}(\xi,t)$ from Eq. (160), we simply substitute
$\eta\to\ln(1+\xi)$ and $t\to(1+\xi)t$ into Eq. (162), obtaining the
relatively simple result

$M_{Q}(\xi,t)=\mathbf{u}^{\sf
T}\overleftarrow{\mathrm{T}}\exp\left[-\xi\int_{0}^{(1+\xi)t}{\rm
d}t^{\prime}\mathsf{\check{R}}(t^{\prime})\right]\mathbf{p}^{\rm ss}.$ (163)

The moments of $Q$ are found from the derivatives of $M_{Q}(\xi,t)$ at
$\xi=0$, as

${\rm
E}[Q^{n}]=(-1)^{n}\left.\frac{\partial^{n}M_{Q}}{\partial\xi^{n}}\right|_{\xi=0}.$
(164)

Carefully applying this formula, we find the mean

${\rm E}[Q]=\int_{0}^{t}{\rm d}t^{\prime}\mathbf{u}^{\sf
T}\mathsf{\check{R}}(t^{\prime})\mathbf{p}^{\rm ss}=t\mathbf{u}^{\sf
T}\mathsf{R}\mathbf{p}^{\rm ss}=\mathcal{A}t,$ (165)

since $\mathbf{u}^{\sf T}\mathsf{L}=\mathsf{L}\mathbf{p}^{\rm ss}=0$ and
$\mathbf{u}^{\sf T}\mathsf{R}\mathbf{p}^{\rm
ss}=\sum_{\mu,\sigma}R_{\mu\sigma}p_{\sigma}^{\rm ss}=\mathcal{A}$ is the
dynamical activity. Similarly, the second moment is

$\displaystyle{\rm E}[Q^{2}]$ $\displaystyle=2\int_{0}^{t}{\rm
d}t^{\prime}\int_{0}^{t^{\prime}}{\rm d}t^{\prime\prime}\mathbf{u}^{\sf
T}\mathsf{R}\mathsf{G}(t^{\prime}-t^{\prime\prime})\mathsf{R}\mathbf{p}^{\rm
ss},$ (166)

and thus the variance ${\rm Var}[Q]={\rm E}[Q^{2}]-(\mathcal{A}t)^{2}$ is

$\displaystyle{\rm Var}[Q]$ $\displaystyle=2\int_{0}^{t}{\rm
d}t^{\prime}\int_{0}^{t^{\prime}}{\rm d}t^{\prime\prime}\mathbf{u}^{\sf
T}\mathsf{R}\left[\mathsf{G}(t^{\prime}-t^{\prime\prime})-\mathbf{p}^{\rm
ss}\mathbf{u}^{\sf T}\right]\mathsf{R}\mathbf{p}^{\rm ss}$
$\displaystyle=2\int_{0}^{t}{\rm d}\tau(t-\tau)\mathbf{u}^{\sf
T}\mathsf{R}\left[\mathsf{G}(\tau)-\mathbf{p}^{\rm ss}\mathbf{u}^{\sf
T}\right]\mathsf{R}\mathbf{p}^{\rm ss}$ $\displaystyle=-2t\mathbf{u}^{\sf
T}\mathsf{R}\mathsf{L}^{+}\mathsf{R}\mathbf{p}^{\rm ss}+\mathcal{O}(t^{0})$
(167)

The second line follows after changing integration variables to
$\tau=t^{\prime}-t^{\prime\prime}$ and $s=(t^{\prime}+t^{\prime\prime})/2$ and
carrying out the trivial integral over $s$, while the final line follows from
extending the upper integration limit to infinity and using Eq. (76).

Notice that $Q$ and $N$ are equal on average, ${\rm E}[N]={\rm E}[Q]$, but $Q$
has a smaller variance, ${\rm Var}[Q]={\rm Var}[N]-{\rm E}[N]$. As a result,
the asymptotic signal-to-noise ratio of $Q$, given by
$\mathcal{S}=\lim_{t\to\infty}\mathcal{A}^{2}/({\rm Var}[Q]/t)$, can exceed
the dynamical activity. This is illustrated by the results in Fig. 6, which
are generated using Eq. (H.3).

##  Appendix I Analytical solution of the Erlang clock

###  I.1 Homogeneous Erlang clock

Here, we provide an analytical solution to the Erlang clock [79, 102] shown in
Fig. 7 of the main text. For readability, we here repeat that this clock for
dimension $d$ is given by a ring network with transition rates
$R_{\mu\sigma}=\Gamma\delta_{\mu,\sigma+1},$ and periodic boundary conditions,
$p_{\mu+d}(t)\equiv p_{\mu}(t)$. The Erlang estimator counts only every $d$th
transition, that is once the system has completed one cycle, the time
estimator is increased by one unit. Formally, this is achieved with the
following weights $w_{\mu\sigma}=d\Gamma^{-1}\delta_{\mu,1}\delta_{\sigma,d}.$
Since only a single type of jump is monitored, the alphabet $\aleph=\\{d\to
1\\}$ has a single element and thus $P(d\to 1|\aleph)=1$, so it follows
immediately from Eq. (43) that the resolution is $\nu=\bar{w}^{-1}=\Gamma/d$.

In the following, we show that the Erlang clock is indeed unbiased, ${\rm
E}[\Theta]=t,$ and moreover that the asymptotic variance is minimal, ${\rm
Var}[\Theta]/t\rightarrow\mathcal{T}^{-1}.$ Finally, we provide an exact
expression for the autocorrelation function shown in Fig. 7 by calculating the
propagator matrix $\mathsf{G}(t)=e^{\mathsf{L}t}$ analytically.

To simplify the notation, we can assume without loss of generality that
$\Gamma=1,$ i.e. parameter time $t$ is expressed in units of $\Gamma.$ With
this, we can directly calculate the expectation value and variance of the
Erlang estimator by using results from Sec. B.4. For the given setup the BLUE
is defined by the weights $w_{\mu+1,\mu}=1,$ which counts all the jumps with
the same weight. Given the cyclic symmetry of our setup, the long-time
variance and expectation value of our estimator are invariant under
transformations $w_{\mu\sigma}^{\prime}=w_{\mu\sigma}+c_{\mu}-c_{\sigma},$ for
an arbitrary vector $c_{\mu},$ as shown in Sec. B.4. By chosing for example
$c_{\mu}=-\mu,$ this symmetry allows us to obtain the Erlang estimator from
the BLUE. We can explicitly calculate
$w^{\prime}_{\mu+1,\mu}=1-(\mu+1)+\mu=0,$ so long as $\mu\leq d-1,$ and for
$\mu=d,$ we have $w^{\prime}_{1d}=1-1+d=d,$ due to cyclicity. For these
indices $\vec{w}^{\prime}$ concides with the Erlang estimator, but for some
other indices $\mu,\sigma,$ we may have $w^{\prime}_{\mu\sigma}\neq 0.$ Since
there are only jumps $\mu\rightarrow\mu+1$, these terms are irrelevant, and
therefore using the results from Sec. B.4 we find that indeed ${\rm
E}[\Theta]=t,$ and ${\rm Var}[\Theta]/t\rightarrow\mathcal{T}^{-1}$ for the
Erlang estimator.

For the next step, we want to calculate the current autocorrelation function
for the Erlang estimator that is shown in Fig. 7. The autocorrelation function
is defined as

$\displaystyle F(t+\tau,t)$ $\displaystyle=\mathrm{Cov}[I(t+\tau),I(t)]$ (168)
$\displaystyle={\rm E}[I(t+\tau)I(t)]-{\rm E}[I(t+\tau)]{\rm E}[I(t)],$ (169)

which in case for the Erlang estimator reduces to determining ${\rm
E}[I(t+\tau)I(t)],$ because we know that in the steady-state limit we have
$\lim_{t\rightarrow\infty}{\rm E}[I(t)]=1$. For values of $\tau\geq 0,$ we can
calculate explicitly

$\displaystyle{\rm E}[I(t+\tau)I(t)]$ $\displaystyle=d^{2}{\rm
E}[I_{1,d}(t+\tau)I_{1,d}(t)]$ (170)
$\displaystyle=d\left[\delta(\tau)+R_{1,d}G_{d,1}(\tau)J_{1,d}(t)\right],$
(171)

having used Eq. (74) for arriving at the second line. In the steady-state
limit $t\rightarrow\infty$, we find $J_{1,d}(t)=1,$ and since
$R_{1,d}=\Gamma=1$ in these units, the autocorrelation function is given by

$\displaystyle\lim_{t\rightarrow\infty}F(t+\tau,t)$
$\displaystyle=d\left(\delta(\tau)+G_{d,1}(\tau)\right)-1.$ (172)

The quantity we still need to determine is the propagator matrix

$\displaystyle\mathsf{G}(t)=e^{\mathsf{L}t}.$ (173)

To compute $\mathsf{G}(t)$, we consider the Poisson process described by the
infinite-dimensional extension of $\mathsf{L}$ as $\mathsf{L}_{\infty}$. More
precisely, we solve a uni-directional hopping process along a semi-infinite
chain of sites described by the probability vector $\mathbf{q}(t)$. Then, the
rate equation $\dot{q}_{\mu}(t)=-q_{\mu}(t)+q_{\mu-1}(t)$ holds for all values
of $\mu\geq 1,$ and we set by definition $q_{0}(t)\equiv 0.$ To recover the
probability vector for the $d$-dimensional Erlang clock, we identify as
equivalent all sites differing by a multiple of $d$ and sum over the
corresponding probabilities:

$\displaystyle p_{\mu}(t)=\sum_{\sigma\geq 0}q_{\mu+\sigma d}(t).$ (174)

We can verify that

$\displaystyle\dot{p}_{\mu}(t)$ $\displaystyle=\sum_{\sigma\geq
0}\dot{q}_{\mu+\sigma d}(t)$ (175) $\displaystyle=\sum_{\sigma\geq
0}\left(q_{\mu+\sigma d-1}(t)-q_{\mu+\sigma d}(t)\right)$ (176)
$\displaystyle=p_{\mu-1}(t)-p_{\mu}(t),$ (177)

which is just what we get from $\dot{\mathbf{p}}(t)=\mathsf{L}\mathbf{p}(t)$
in the $d$-dimensional case. This proves that the identification in Eq. (174)
is consistent with the two equations of motion for $\mathbf{q}$ and
$\mathbf{p}$.

Now, to solve the equation of motion for $\mathbf{q}(t)$ we introduce the
moment generating function $\rho(t,\chi)=\sum_{\mu\geq 0}p_{\mu}(t){\rm
e}^{{\rm i}\mu\chi}$, such that the following two equations are equivalent:

$\displaystyle\dot{\mathbf{q}}(t)=\mathsf{L}_{\infty}\mathbf{q}(t)\Leftrightarrow\dot{\rho}(t,\chi)=({\rm
e}^{{\rm i}\chi}-1)\rho(t,\chi).$ (178)

The one on the RHS we can trivially integrate and we find

$\displaystyle\rho(t,\chi)$ $\displaystyle={\rm e}^{t({\rm e}^{{\rm
i}\chi}-1)}\rho(0,\chi)$ (179) $\displaystyle=\sum_{\mu,\sigma\geq
0}\frac{t^{\mu}}{\mu!}{\rm e}^{{\rm i}\mu\chi}{\rm e}^{-t}q_{\sigma}(t){\rm
e}^{{\rm i}\sigma\chi}$ (180)

which allows us to analytically calculate
$\mathsf{G}_{\infty}(t)=\exp(\mathsf{L}_{\infty}t)$. Given the initial
condition $\rho^{(\sigma)}(0,\chi)={\rm e}^{{\rm i}\sigma\chi},$ we can
determine

$\displaystyle[\mathsf{G}_{\infty}(t)]_{\mu\sigma}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}{\rm d}\chi\,{\rm e}^{-{\rm
i}\mu\chi}\rho^{(\sigma)}(t,\chi)$ (181)
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}{\rm d}\chi\,\sum_{\nu\geq
0}\frac{t^{\nu}}{\nu!}{\rm e}^{-t}{\rm e}^{{\rm i}\chi(\sigma+\nu-\mu)}$ (182)
$\displaystyle=\frac{t^{\mu-\sigma}}{(\mu-\sigma)!}{\rm e}^{-t}.$ (183)

Non-surprisingly, we recover the Poisson-distribution, defined by the rate
equation for $\mathbf{q}(t).$

Finally, we can calculate the propagator matrix $\mathsf{G}(t)$ for the
$d$-dimensional system by using Eq. (174) to get

$\displaystyle G_{\mu\sigma}(t)$ $\displaystyle=\sum_{\nu\geq
0}[\mathsf{G}_{\infty}(t)]_{\mu+\nu d,\sigma}$ (184)
$\displaystyle=\sum_{\nu\geq 0}\frac{t^{\mu-\sigma+\nu d}}{(\mu-\sigma+\nu
d)!}{\rm e}^{-t}.$ (185)

Given this result, we can calculate $G_{d,1}(t)$ which we need for the
autocorrelation function,

$\displaystyle\lim_{t\rightarrow\infty}F(t+\tau,t)=d\left(\delta(\tau)+\sum_{\mu\geq
0}\frac{\tau^{d-1+\mu d}}{(d-1+\mu d)!}{\rm e}^{-\tau}\right)-1,$ (186)

the result of which is plotted in Fig. 7. Note that from this we can obtain
the short-time variance, as

${\rm Var}[\Theta]=\int_{0}^{t}{\rm d}t_{1}\int_{0}^{t}{\rm
d}t_{2}F(t_{1},t_{2})=dt+\mathcal{O}(t^{2}),$ (187)

or, restoring units, we have $\lim_{t\to 0}{\rm Var}[\Theta_{\rm
Erl}]/t=d/\Gamma$, which is $d$ times larger than the variance of the BLUE,
${\rm Var}[\Theta_{\rm BLUE}]/t=\mathcal{T}=\Gamma^{-1}$.

###  I.2 Inhomogeneous Erlang clock

One can also modify the Erlang clock to accommodate inhomogeneous rates,
$R_{\mu\sigma}=\Gamma_{\sigma}\delta_{\mu,\sigma+1}$. As before, only
clockwise motion around the ring is allowed, but now the escape rate
$\Gamma_{\sigma}$ can be different for each site. While a complete solution
for the dynamics is cumbersome in this case, we can still obtain simple
expressions for the asymptotic precision and resolution of the Erlang
estimator, which counts only a single transition.

####  I.2.1 Two-state system

Let us start with the simplest case of $d=2$, whose dynamics was already
solved exactly in Appendix G. Here we recap the steady-state probabilities
$p_{1}^{\rm ss}=\Gamma_{2}/(\Gamma_{1}+\Gamma_{2})$ and $p_{2}^{\rm
ss}=\Gamma_{1}/(\Gamma_{1}+\Gamma_{2})$, so that the mean residual time is

$\mathcal{T}=\frac{\Gamma_{1}^{2}+\Gamma_{2}^{2}}{\Gamma_{1}\Gamma_{2}(\Gamma_{1}+\Gamma_{2})}.$
(188)

The Erlang estimator is the unbiased counting estimator that counts only the
transition $2\to 1$, i.e. $w_{12}=0$ and

$w_{12}=J_{12}^{-1}=\frac{\Gamma_{1}+\Gamma_{2}}{\Gamma_{1}\Gamma_{2}}.$ (189)

Since only a single transition is counted, its resolution is simply the
inverse of this increment:

$\nu=\frac{\Gamma_{1}\Gamma_{2}}{\Gamma_{1}+\Gamma_{2}}.$ (190)

To show that this estimator has minimal variance in the long-time limit, we
use the fact that the ticks of the Erlang estimator are a renewal process
[101]: the state of the system is reset to the same state, $\sigma=1$, with
the same ensuing dynamics after every tick. A renewal process is fully
specified by the distribution of waiting times between ticks,

$W_{d}(t)=-\frac{{\rm d}P_{{\rm no},d}}{{\rm d}t},$ (191)

where $P_{{\rm no},d}(t)$ is the probability that no tick occurs up to time
$t$ given that a tick was detected at $t=0$. For the Erlang estimator with
$d=2$, this is equivalent to the probability that at most one jump has
occurred after the system was set to state $\sigma=1$, so we find

$P_{{\rm no},2}(t)={\rm e}^{-\Gamma_{1}t}+\int_{0}^{t}{\rm d}t_{1}{\rm
e}^{-\Gamma_{2}(t-t_{1})}\Gamma_{1}{\rm e}^{-\Gamma_{1}t_{1}},$ (192)

where the first term is the probability of $N=0$ jumps and the second term is
the probability for $N=1$ (cf. Eq. (53)). Plugging this into Eq. (191) yields
the tick distribution

$\displaystyle W_{2}(t)$ $\displaystyle=\int_{0}^{t}{\rm d}t_{1}\Gamma_{2}{\rm
e}^{-\Gamma_{2}(t-t_{1})}\Gamma_{1}{\rm e}^{-\Gamma_{1}t_{1}}$ (193)
$\displaystyle=\frac{\Gamma_{1}\Gamma_{2}\left({\rm e}^{-\Gamma_{2}t}-{\rm
e}^{-\Gamma_{1}t}\right)}{\Gamma_{1}-\Gamma_{2}}.$ (194)

The mean and variance of the tick waiting times are thus found to be

$\displaystyle{\rm E}[t_{2}]=\int_{0}^{\infty}{\rm
d}t\,W_{2}(t)t=\frac{\Gamma_{1}+\Gamma_{2}}{\Gamma_{1}\Gamma_{2}}=\nu^{-1},$
(195) $\displaystyle{\rm Var}[t_{2}]=\int_{0}^{\infty}{\rm
d}t\,W_{2}(t)\left(t-{\rm
E}[t_{2}]\right)^{2}=\frac{\Gamma_{1}^{2}+\Gamma_{2}^{2}}{\Gamma_{1}^{2}\Gamma_{2}^{2}}.$
(196)

Now, the Erlang estimator itself is $\Theta_{\rm Erl}={\rm E}[t_{2}]N_{2}$,
where $N_{2}$ is the number of ticks. Since $N_{2}$ is renewal, its long-time
diffusion coefficient can be shown to be (e.g. see Sec.VI.B.2 of Ref. [44])

$\lim_{t\to\infty}\frac{{\rm d}}{{\rm d}t}{\rm Var}[N_{2}]=\frac{{\rm
Var}[t_{2}]}{{\rm E}[t_{2}]^{3}}.$ (197)

It follows that the diffusion coefficient for the Erlang estimator itself is
$D={\rm Var}[t_{2}]/{\rm E}[t_{2}]=\mathcal{T}$. This proves that the Erlang
estimator is an unbiased counting estimator with minimal variance, i.e. it
constitutes a solution to Eq. (28). Its resolution, on the other hand, is
given by $\nu={\rm E}[t_{2}]^{-1}=\mathcal{A}/2$, where the dynamical activity
is $\mathcal{A}=2\Gamma_{1}\Gamma_{2}/(\Gamma_{1}+\Gamma_{2})$.

####  I.2.2 Arbitrary number of states

To extend this to larger values of $d$, we first find the steady-state
probability distribution. Since the equation of motion is
$\partial_{t}p_{\sigma}=\Gamma_{\sigma-1}p_{\sigma-1}-\Gamma_{\sigma}p_{\sigma}$,
the steady-state solution obeys the recursion relation

$p_{\sigma}^{\rm
ss}=\frac{\Gamma_{\sigma-1}}{\Gamma_{\sigma}}p_{\sigma-1}^{\rm ss}.$ (198)

The solution is $p_{\sigma}^{\rm ss}={\rm const.}\times\Gamma_{\sigma}^{-1}$,
and taking into account normalisation, we find

$p_{\sigma}^{\rm ss}=\frac{\mathcal{A}}{d\Gamma_{\sigma}},$ (199)

where the dynamical activity is given by

$\mathcal{A}=\sum_{\sigma}p_{\sigma}^{\rm
ss}\Gamma_{\sigma}=\left(\sum_{\sigma=1}^{d}\frac{1}{d\Gamma_{\sigma}}\right)^{-1}.$
(200)

We also find the mean residual time

$\mathcal{T}=\sum_{\sigma}\frac{p_{\sigma}^{\rm
ss}}{\Gamma_{\sigma}}=\sum_{\sigma=1}^{d}\frac{\mathcal{A}}{d\Gamma_{\sigma}^{2}}.\\\
$ (201)

Now, we define the Erlang estimator by setting all weights $w_{\mu\sigma}=0$
except for

$w_{1d}=J_{1d}^{-1}.$ (202)

This choice guarantees that the estimator is unbiased, since
$\vec{w}\cdot\vec{J}=w_{1d}J_{1d}=1.$ Only one jump is counted, therefore the
resolution is simply the inverse of Eq. (202). More precisely, we have

$\nu=\frac{\mathcal{A}}{d},$ (203)

where we have used $J_{1d}=\Gamma_{d}p_{d}^{\rm ss}$ together with Eq. (199).
Therefore, the Erlang estimator has resolution $d$ times smaller than the
BLUE, for any set of rates $\\{\Gamma_{\sigma}\\}$ and dimension $d$.

As before, we are dealing with a renewal process since the state is reset to
$\sigma=1$ after each tick. We therefore need only find the tick waiting-time
distribution to obtain the asymptotic statistics of the estimator. For
arbitrary $d$, Eq. (193) generalises to

$W_{d}(t)=\int_{0}^{t}{\rm d}t_{d-1}\cdots\int_{0}^{t_{2}}{\rm
d}t_{1}\prod_{\sigma=1}^{d}W(t_{\sigma}-t_{\sigma-1}|\sigma),$ (204)

where $W(\tau|\sigma)=\Gamma_{\sigma}{\rm e}^{-\Gamma_{\sigma}\tau}$ is the
distribution of waiting times preceding a jump from state $\sigma$, and we
define $t_{d}\equiv t$ and $t_{0}\equiv 0$. This expresses the fact that the
total time between ticks is given by the sum of $d$ independent waiting times,
corresponding to the $d$ sequential jumps $\sigma\to\sigma+1$ that take the
system once around the clock. The probability of the total waiting time is
thus a convolution of the individual distributions, as in Eq. (204).

By the convolution theorem, the Laplace transform of the tick distribution
(204) is

$\tilde{W}_{d}(z)=\int_{0}^{\infty}{\rm d}t\,{\rm
e}^{-zt}W_{d}(t)=\prod_{\sigma=1}^{d}\frac{\Gamma_{\sigma}}{z+\Gamma_{\sigma}},$
(205)

where $\tilde{W}(z|\sigma)=\Gamma_{\sigma}/(z+\Gamma_{\sigma})$ is the Laplace
transform of the waiting-time distribution for a single jump. To find the mean
and variance of the total waiting time between ticks, it is sufficient to note
that $\tilde{W}_{d}(z)$ is the moment-generating function for the tick waiting
times, i.e.

${\rm E}[t_{d}^{n}]=\int_{0}^{\infty}{\rm
d}t\,W_{d}(t)t^{n}=(-1)^{n}\left.\frac{\partial^{n}\tilde{W}_{d}(z)}{\partial
z^{n}}\right|_{z=0}.$ (206)

The corresponding cumulants can thus be found by taking derivatives of
$\ln\tilde{W}_{d}(z)$ at $z=0$, viz.

$\displaystyle{\rm
E}[t_{d}]=-\partial_{z}\ln\tilde{W}_{d}(z)|_{z=0}=\sum_{\sigma=1}^{d}\Gamma_{\sigma}^{-1},$
(207) $\displaystyle{\rm
Var}[t_{d}]=\partial^{2}_{z}\ln\tilde{W}_{d}(z)|_{z=0}=\sum_{\sigma=1}^{d}\Gamma_{\sigma}^{-2}.$
(208)

We recognise immediately that the mean time between ticks is ${\rm
E}[t_{d}]=d/\mathcal{A}=\nu^{-1}$ (cf. Eq. (203)), as expected. Using Eqs.
(200)–(201) and Eqs. (207)–(208), we obtain the diffusion coefficient (cf. the
discussion around Eq. (197))

$D=\frac{{\rm Var}[t_{d}]}{{\rm E}[t_{d}]}=\mathcal{T}.$ (209)

This proves the results for the Erlang estimator summarised in Table 2. It is
notable that these results do not depend on the specific value of
$\Gamma_{d}$, only on aggregate properties of the set $\\{\Gamma_{\sigma}\\}$.
This shows that there is nothing special about the transition $d\to 1$ even in
the inhomogeneous case: an unbiased Erlang estimator can be defined for any
other pair of adjacent sites, yielding the same asymptotic precision and
resolution.

##  Appendix J Allan variance

In this Appendix, we compute the Allan variance for a stationary counting
observable. The Allan variance is defined in Eq. (49), which we can also write
as

$\sigma_{y}^{2}(T)=\frac{1}{2}{\rm
E}\left[\Big{(}\bar{y}_{T}(t+T)-\bar{y}_{T}(t)\Big{)}^{2}\right],$ (210)

where the average fractional frequency error is

$\bar{y}_{T}(t)=\frac{1}{T}\int_{t}^{t+T}{\rm
d}t^{\prime}\,\left[I(t^{\prime})-1\right].$ (211)

Since $y_{T}(t)$ is a stationary, zero-mean process, we have

$\displaystyle\sigma_{y}^{2}(T)$ $\displaystyle=\frac{1}{2}\left\\{{\rm
Var}[\bar{y}_{T}(t+T)]+{\rm Var}[\bar{y}_{T}(t)]\right.$
$\displaystyle\qquad+\left.2\,{\rm
Cov}[\bar{y}_{T}(t+T),\bar{y}_{T}(t)]\right\\}$ $\displaystyle={\rm
Var}[\bar{y}_{T}(0)]+{\rm Cov}[\bar{y}_{T}(T),\bar{y}_{T}(0)].$ (212)

Upon plugging in Eq. (211) for $y_{T}(t)$, this yields

$\displaystyle\sigma_{y}^{2}(T)=$ $\displaystyle\frac{{\rm
Var}[\Theta|t=T]}{T^{2}}+\frac{1}{T^{2}}\int_{T}^{2T}\\!\\!{\rm
d}t_{1}\int_{0}^{T}\\!\\!{\rm d}t_{2}\,F(t_{1}-t_{2}),$ (213)

with $F(t_{1}-t_{2})={\rm Cov}[I(t_{1}),I(t_{2})]$. The variance in the first
term is expressed in terms of the autocorrelation function as

$\displaystyle{\rm Var}[\Theta|t=T]$ $\displaystyle=\int_{0}^{T}\\!\\!{\rm
d}t_{1}\int_{0}^{T}\\!\\!{\rm d}t_{2}\,F(t_{1}-t_{2})$
$\displaystyle=2\int_{0}^{T}\\!{\rm d}\tau\,(T-\tau)F(\tau),$ (214)

where we changed integration variables to $\tau=t_{1}-t_{2}$ and
$s=(t_{1}+t_{2})/2$, and carried out the trivial integral over $s$. Similarly,
the second term in Eq. (213) can be written as

${\rm Cov}[\bar{y}_{T}(T),\bar{y}_{T}(0)]=\int_{0}^{T}{\rm d}\tau\,\tau
F(\tau)+\int_{T}^{2T}{\rm d}\tau\,(2T-\tau)F(\tau).$ (215)

Now, for $\tau\geq 0$ we can use Eq. (82) to write the autocorrelator as

$\displaystyle F(\tau)$ $\displaystyle=K\delta(\tau)+\sum_{k=0}^{d^{2}}{\rm
e}^{\lambda_{k}\tau}\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{\Pi}_{k}\tilde{\mathsf{J}}\mathbf{p}^{\rm
ss}-J^{2}$ $\displaystyle=K\delta(\tau)+\sum_{k\neq 0}{\rm
e}^{\lambda_{k}\tau}\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{\Pi}_{k}\tilde{\mathsf{J}}\mathbf{p}^{\rm ss},$
(216)

where $\mathsf{\Pi}_{k}=\mathbf{r}_{k}\mathbf{l}^{\sf T}_{k}$ is the projector
onto $\mathbf{r}_{k}$, the right eigenvector of $\mathsf{L}$ with eigenvalue
$\lambda_{k}$, with $\mathbf{l}_{k}^{\sf T}$ the corresponding left (row)
eigenvector [44]. The left and right eigenvectors form a biorthonormal set,
such that $\mathbf{l}_{k}^{\sf
T}\mathbf{r}_{k^{\prime}}=\delta_{kk^{\prime}}$. The second line of Eq. (J)
follows from the fact that $\lambda_{0}=0$, with the steady state
$\mathbf{p}^{\rm ss}=\mathbf{r}_{0}$ and the constant row vector
$\mathbf{u}^{\sf T}=\mathbf{l}^{\sf T}_{0}$ as the corresponding right and
left eigenvectors. We therefore have $\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{\Pi}_{0}\tilde{\mathsf{J}}\mathbf{p}^{\rm
ss}=(\mathbf{u}^{\sf T}\tilde{\mathsf{J}}\mathbf{p}^{\rm ss})^{2}=J^{2}$ from
Eq. (81).

Using Eq. (J) in Eq. (J) and performing the integral, we have

$\displaystyle{\rm Var}[\Theta|t=T]$ $\displaystyle=KT+2\sum_{k\neq
0}\frac{{\rm e}^{\lambda_{k}T}-1-T\lambda_{k}}{\lambda^{2}_{k}}\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{\Pi}_{k}\tilde{\mathsf{J}}\mathbf{p}^{\rm ss}.$
(217)

Now, using the fact that $\\{\mathsf{\Pi}_{k}\\}$ is an orthonormal, complete
set of projectors,
$\mathsf{\Pi}_{k}\mathsf{\Pi}_{k^{\prime}}=\delta_{k,k^{\prime}}\mathsf{\Pi}_{k}$
and $\sum_{k}\mathsf{\Pi}_{k}=\mathsf{1}$, and that $\sum_{k\neq
0}\mathsf{\Pi}_{k}/\lambda_{k}=\mathsf{L}^{+}$ is the Drazin inverse [44], we
can write

$\displaystyle{\rm Var}[\Theta|t=T]$ $\displaystyle=DT+2\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{L}^{+}\left[\mathsf{G}(T)-\mathsf{1}\right]\mathsf{L}^{+}\tilde{\mathsf{J}}\mathbf{p}^{\rm
ss},$ (218)

where we have used Eq. (83) to collect terms equal to the diffusion
coefficient, $D$. A similar calculation can be performed for Eq. (215),
yielding a relatively compact expression for the Allan variance,

$\sigma_{y}^{2}(T)=\frac{D}{T}+\frac{1}{T^{2}}\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{L}^{+}\left\\{2\left[\mathsf{G}(T)-\mathsf{1}\right]-\left[\mathsf{G}(T)-\mathsf{1}\right]^{2}\right\\}\mathsf{L}^{+}\tilde{\mathsf{J}}\mathbf{p}^{\rm
ss},$ (219)

which we use to generate the data in Fig. 8.

At long times, the second term of Eq. (219) is subleading with respect to the
first, and we recover simply

$\sigma^{2}_{y}(T)=\frac{D}{T}+\mathcal{O}(T^{-2}).$ (220)

For short times, a power expansion of $\mathsf{G}(T)={\rm e}^{\mathsf{L}T}$
gives

$\displaystyle\sigma_{y}^{2}(T)$
$\displaystyle=\frac{D}{T}+\frac{1}{T}\mathbf{u}^{\sf
T}\tilde{\mathsf{J}}\mathsf{L}^{+}\mathsf{L}\mathsf{L}^{+}\tilde{\mathsf{J}}\mathbf{p}^{\rm
ss}+\mathcal{O}(T)$ $\displaystyle=\frac{K}{T}+\mathcal{O}(T)$ (221)

where we again used Eq. (83). Thus, we confirm the long- and short-time limits
quoted in Eq. (50). While the long-time Allan variance (220) is clearly
minimised by the BLUE, which has minimal $D$, minimising the short-time Allan
variance requires extremising
$K=\sum_{\mu,\sigma}w_{\mu\sigma}^{2}J_{\mu\sigma}$. This is achieved by
choosing uniform weights $w_{\mu\sigma}=\rm const.$ which, together with the
unbiased constraint $J=\vec{w}\cdot\vec{J}=1$, implies that
$w_{\mu\sigma}=\mathcal{A}^{-1}$.

The Allan variance simplifies drastically for the BLUE defined by Eq. (29), as
a consequence of its autocorrelation function
$F(\tau)=\mathcal{T}\delta(\tau)$. Plugging this into Eq. (213), the second
term vanishes while the variance becomes ${\rm Var}[\Theta|t]=Dt$ (cf. Eq.
(31)). We therefore conclude that, for the BLUE, $\sigma_{y}^{2}(T)=D/T$
exactly for all $T$.

## References

  * Erker  _et al._ [2017] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?, Phys. Rev. X 7, 031022 (2017).
  * Woods  _et al._ [2019] M. P. Woods, R. Silva, and J. Oppenheim, Autonomous Quantum Machines and Finite-Sized Clocks, Ann. Henri Poincaré 20, 125 (2019).
  * Milburn [2020] G. J. Milburn, The thermodynamics of clocks, Contemp. Phys. 61, 69 (2020).
  * Schwarzhans  _et al._ [2021] E. Schwarzhans, M. P. E. Lock, P. Erker, N. Friis, and M. Huber, Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics, Phys. Rev. X 11, 011046 (2021).
  * Woods [2021] M. P. Woods, Autonomous Ticking Clocks from Axiomatic Principles, Quantum 5, 381 (2021).
  * Woods  _et al._ [2022] M. P. Woods, R. Silva, G. Pütz, S. Stupar, and R. Renner, Quantum Clocks are More Accurate Than Classical Ones, PRX Quantum 3, 010319 (2022).
  * Meier  _et al._ [2023] F. Meier, E. Schwarzhans, P. Erker, and M. Huber, Fundamental Accuracy-Resolution Trade-Off for Timekeeping Devices, Phys. Rev. Lett. 131, 220201 (2023).
  * Pearson  _et al._ [2021] A. N. Pearson, Y. Guryanova, P. Erker, E. A. Laird, G. A. D. Briggs, M. Huber, and N. Ares, Measuring the Thermodynamic Cost of Timekeeping, Phys. Rev. X 11, 021029 (2021).
  * He  _et al._ [2023] X. He, P. Pakkiam, A. A. Gangat, M. J. Kewming, G. J. Milburn, and A. Fedorov, Effect of measurement backaction on quantum clock precision studied with a superconducting circuit, Phys. Rev. Appl. 20, 034038 (2023).
  * Culhane  _et al._ [2024] O. Culhane, M. J. Kewming, A. Silva, J. Goold, and M. T. Mitchison, Powering an autonomous clock with quantum electromechanics, New J. Phys. 26, 023047 (2024).
  * Gopal  _et al._ [2024] A. Gopal, M. Esposito, and N. Freitas, Thermodynamic cost of precise timekeeping in an electronic underdamped clock, Phys. Rev. B 109, 085421 (2024).
  * Barato and Seifert [2016] A. C. Barato and U. Seifert, Cost and Precision of Brownian Clocks, Phys. Rev. X 6, 041053 (2016).
  * Barato and Seifert [2017] A. C. Barato and U. Seifert, Coherence of biochemical oscillations is bounded by driving force and network topology, Phys. Rev. E 95, 062409 (2017).
  * Marsland  _et al._ [2019] R. Marsland, W. Cui, and J. M. Horowitz, The thermodynamic uncertainty relation in biochemical oscillations, J. R. Soc. Interface. 16, 20190098 (2019).
  * Silva  _et al._ [2023] R. Silva, N. Nurgalieva, and H. Wilming, Ticking clocks in quantum theory (2023), arXiv:2306.01829 [quant-ph] .
  * Barato and Seifert [2015] A. C. Barato and U. Seifert, Thermodynamic Uncertainty Relation for Biomolecular Processes, Phys. Rev. Lett. 114, 158101 (2015).
  * Gingrich  _et al._ [2016] T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation Bounds All Steady-State Current Fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  * Garrahan [2017] J. P. Garrahan, Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables, Phys. Rev. E 95, 032134 (2017).
  * Terlizzi and Baiesi [2018] I. D. Terlizzi and M. Baiesi, Kinetic uncertainty relation, J. Phys. A: Math. Theor. 52, 02LT03 (2018).
  * Shiraishi  _et al._ [2016] N. Shiraishi, K. Saito, and H. Tasaki, Universal trade-off relation between power and efficiency for heat engines, Phys. Rev. Lett. 117, 190601 (2016).
  * Pietzonka and Seifert [2018] P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Phys. Rev. Lett. 120, 190602 (2018).
  * Miller  _et al._ [2021] H. J. D. Miller, M. H. Mohammady, M. Perarnau-Llobet, and G. Guarnieri, Thermodynamic uncertainty relation in slowly driven quantum heat engines, Phys. Rev. Lett. 126, 210603 (2021).
  * Li  _et al._ [2019] J. Li, J. M. Horowitz, T. R. Gingrich, and N. Fakhri, Quantifying dissipation using fluctuating currents, Nat. Commun. 10, 1666 (2019).
  * Manikandan  _et al._ [2020] S. K. Manikandan, D. Gupta, and S. Krishnamurthy, Inferring entropy production from short experiments, Phys. Rev. Lett. 124, 120603 (2020).
  * Van Vu  _et al._ [2020] T. Van Vu, V. T. Vo, and Y. Hasegawa, Entropy production estimation with optimal current, Phys. Rev. E 101, 042138 (2020).
  * Pietzonka  _et al._ [2017] P. Pietzonka, F. Ritort, and U. Seifert, Finite-time generalization of the thermodynamic uncertainty relation, Phys. Rev. E 96, 012101 (2017).
  * Horowitz and Gingrich [2017] J. M. Horowitz and T. R. Gingrich, Proof of the finite-time thermodynamic uncertainty relation for steady-state currents, Phys. Rev. E 96, 020103 (2017).
  * Gingrich and Horowitz [2017] T. R. Gingrich and J. M. Horowitz, Fundamental bounds on first passage time fluctuations for currents, Phys. Rev. Lett. 119, 170601 (2017).
  * Hiura and Sasa [2021] K. Hiura and S.-i. Sasa, Kinetic uncertainty relation on first-passage time for accumulated current, Phys. Rev. E 103, L050103 (2021).
  * Proesmans and den Broeck [2017] K. Proesmans and C. V. den Broeck, Discrete-time thermodynamic uncertainty relation, EPL 119, 20001 (2017).
  * Koyuk and Seifert [2020] T. Koyuk and U. Seifert, Thermodynamic Uncertainty Relation for Time-Dependent Driving, Phys. Rev. Lett. 125, 260604 (2020).
  * Brandner  _et al._ [2018] K. Brandner, T. Hanazato, and K. Saito, Thermodynamic bounds on precision in ballistic multiterminal transport, Phys. Rev. Lett. 120, 090601 (2018).
  * Macieszczak  _et al._ [2018] K. Macieszczak, K. Brandner, and J. P. Garrahan, Unified thermodynamic uncertainty relations in linear response, Phys. Rev. Lett. 121, 130601 (2018).
  * Guarnieri  _et al._ [2019] G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Thermodynamics of precision in quantum nonequilibrium steady states, Phys. Rev. Res. 1, 033021 (2019).
  * Hasegawa [2020] Y. Hasegawa, Quantum thermodynamic uncertainty relation for continuous measurement, Phys. Rev. Lett. 125, 050601 (2020).
  * Van Vu and Saito [2022] T. Van Vu and K. Saito, Thermodynamics of precision in Markovian open quantum dynamics, Phys. Rev. Lett. 128, 140602 (2022).
  * Salazar [2024] D. S. P. Salazar, Thermodynamic uncertainty relation for quantum entropy production (2024), arXiv:2404.18163 [quant-ph] .
  * Potts and Samuelsson [2019] P. P. Potts and P. Samuelsson, Thermodynamic uncertainty relations including measurement and feedback, Phys. Rev. E 100, 052137 (2019).
  * Timpanaro  _et al._ [2019] A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from exchange fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019).
  * Hasegawa and Van Vu [2019a] Y. Hasegawa and T. Van Vu, Fluctuation theorem uncertainty relation, Phys. Rev. Lett. 123, 110602 (2019a).
  * Ray  _et al._ [2023] K. J. Ray, A. B. Boyd, G. Guarnieri, and J. P. Crutchfield, Thermodynamic uncertainty theorem, Phys. Rev. E 108, 054126 (2023).
  * Seifert [2019] U. Seifert, From Stochastic Thermodynamics to Thermodynamic Inference, Annu. Rev. Condens. Matter Phys. 10, 171 (2019).
  * Horowitz and Gingrich [2020] J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  * Landi  _et al._ [2024] G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quantum 5, 020201 (2024).
  * Landi and Paternostro [2021] G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  * Maes [2020] C. Maes, Frenesy: Time-symmetric dynamical activity in nonequilibria, Phys. Rep. 850, 1 (2020).
  * Stein and Dattero [1985] W. E. Stein and R. Dattero, Sampling Bias and the Inspection Paradox, Math. Mag. 58, 96 (1985).
  * Pal  _et al._ [2022] A. Pal, S. Kostinski, and S. Reuveni, The inspection paradox in stochastic resetting, J. Phys. A: Math. Theor. 55, 021001 (2022).
  * Kay [2013] S. M. Kay,  _Fundamentals of Statistical Signal Processing. 1: Estimation Theory_ (Prentice Hall PTR, Upper Saddle River, NJ, 2013).
  * Ito [2018] S. Ito, Stochastic thermodynamic interpretation of information geometry, Phys. Rev. Lett. 121, 030605 (2018).
  * Ito and Dechant [2020] S. Ito and A. Dechant, Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound, Phys. Rev. X 10, 021056 (2020).
  * Nicholson  _et al._ [2020] S. B. Nicholson, L. P. García-Pintos, A. del Campo, and J. R. Green, Time–information uncertainty relations in thermodynamics, Nat. Phys. 16, 1211 (2020).
  * Gardiner [2009] C. W. Gardiner,  _Stochastic Methods: A Handbook for the Natural and Social Sciences_ , 4th ed., Springer Series in Synergetics No. 13 (Springer, Berlin Heidelberg, 2009).
  * Schmiedl and Seifert [2007] T. Schmiedl and U. Seifert, Stochastic thermodynamics of chemical reaction networks, J. Chem. Phys. 126, 044101 (2007).
  * Goldbeter and Berridge [1996] A. Goldbeter and M. J. Berridge,  _Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour_, 1st ed. (Cambridge University Press, Cambridge, 1996).
  * Breuer and Petruccione [2007] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_, 1st ed. (Oxford University Press, Oxford, 2007).
  * McFadden [1962] J. A. McFadden, On the Lengths of Intervals in a Stationary Point Process, J. R. Stat. Soc. B 24, 364 (1962).
  * Note [1] Technically, this assumes that the total time $t$ is large enough to avoid boundary effects due to the constraint $\sum@\slimits@_{j=0}^{N}\tau_{j}=t$.
  * Landi [2023] G. T. Landi, Patterns in the jump-channel statistics of open quantum systems (2023), arXiv:2305.07957 [quant-ph] .
  * Note [2] This can be proved, for example, using the Cauchy-Schwarz inequality $\mathbf{a}^{\sf T}\mathbf{b}\leq\lVert\mathbf{a}\rVert\cdot\lVert\mathbf{b}\rVert$, with $a_{\sigma}=\sqrt{p^{\rm ss}_{\sigma}\Gamma_{\sigma}}$ and $b_{\sigma}=\sqrt{p^{\rm ss}_{\sigma}/\Gamma_{\sigma}}$. The necessary and sufficient condition for equality is $a_{\sigma}=\mathrm{const.}\times b_{\sigma}$, which implies $\Gamma_{\sigma}=\mathrm{const.}$ whenever $p^{\rm ss}_{\sigma}\neq 0$.
  * Smiga  _et al._ [2023] J. A. Smiga, M. Radaelli, F. C. Binder, and G. T. Landi, Stochastic metrology and the empirical distribution, Phys. Rev. Res. 5, 033150 (2023).
  * Touchette [2011] H. Touchette, A basic introduction to large deviations: Theory, applications, simulations, in  _Modern Computational Science 11: Lecture Notes from the 3rd International Oldenburg Summer School_ (BIS-Verlag der Carl von Ossietzky Universitat Oldenburg, 2011).
  * Note [3] This should be distinguished from the empirical measure considered in large deviation theory [18], which gives the fraction of total time spent in state $\sigma$. The empirical measure tends to the steady-state distribution $p_{\sigma}^{\rm ss}$ when averaged over a long observation time, while the empirical distribution tends to the jump steady state; see Appendix C for an illustration.
  * Note [4] Note that we use the asymptotic (big-$\mathcal{O}$) notation following Knuth [103] and by $X(t)\sim Y(t)$ we mean that $X(t)=\mathcal{O}(Y(t))$ and $Y(t)=\mathcal{O}(X(t))$, i.e. $X$ is proportional to $Y$ asymptotically for large $t$.
  * Radaelli  _et al._ [2023] M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, Fisher information of correlated stochastic processes, New J. Phys. 25, 053037 (2023).
  * Busiello and Pigolotti [2019] D. M. Busiello and S. Pigolotti, Hyperaccurate currents in stochastic thermodynamics, Phys. Rev. E 100, 060102 (2019).
  * Timpanaro  _et al._ [2023] A. M. Timpanaro, G. Guarnieri, and G. T. Landi, Hyperaccurate thermoelectric currents, Phys. Rev. B 107, 115432 (2023).
  * Lang  _et al._ [2014] A. H. Lang, C. K. Fisher, T. Mora, and P. Mehta, Thermodynamics of statistical inference by cells, Phys. Rev. Lett. 113, 148103 (2014).
  * Harvey  _et al._ [2023] S. E. Harvey, S. Lahiri, and S. Ganguli, Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing, Phys. Rev. E 108, 014403 (2023).
  * Dechant and Sasa [2020] A. Dechant and S.-i. Sasa, Fluctuation–response inequality out of equilibrium, Proc. Nat. Acad. Sci. 117, 6430 (2020).
  * Shiraishi [2021] N. Shiraishi, Optimal Thermodynamic Uncertainty Relation in Markov Jump Processes, J. Stat. Phys. 185, 19 (2021).
  * Dechant [2018] A. Dechant, Multidimensional thermodynamic uncertainty relations, J. Phys. A: Math. Theor. 52, 035001 (2018).
  * Hasegawa and Van Vu [2019b] Y. Hasegawa and T. Van Vu, Uncertainty relations in stochastic processes: An information inequality approach, Phys. Rev. E 99, 062126 (2019b).
  * Vo  _et al._ [2022] V. T. Vo, T. V. Vu, and Y. Hasegawa, Unified thermodynamic–kinetic uncertainty relation, J. Phys. A: Math. Theor. 55, 405004 (2022).
  * Zamir [1998] R. Zamir, A proof of the Fisher information inequality via a data processing argument, IEEE Trans. Inf. Theory 44, 1246 (1998).
  * Xuereb  _et al._ [2023] J. Xuereb, P. Erker, F. Meier, M. T. Mitchison, and M. Huber, Impact of imperfect timekeeping on quantum control, Phys. Rev. Lett. 131, 160204 (2023).
  * Allan [1966] D. Allan, Statistics of atomic frequency standards, Proc. IEEE 54, 221 (1966).
  * Bothwell  _et al._ [2022] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, and J. Ye, Resolving the gravitational redshift across a millimetre-scale atomic sample, Nature 602, 420 (2022).
  * Erlang [1917] A. K. Erlang, Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Post Office Electrical Engineer’s Journal 10, 189 (1917).
  * Meier  _et al._ [2024] F. Meier, M. Huber, P. Erker, and J. Xuereb, Autonomous Quantum Processing Unit: What does it take to construct a self-contained model for quantum computation? (2024), arXiv:2402.00111 [quant-ph] .
  * Moffitt and Bustamante [2014] J. R. Moffitt and C. Bustamante, Extracting signal from noise: Kinetic mechanisms from a Michaelis–Menten-like expression for enzymatic fluctuations, FEBS J. 281, 498 (2014).
  * Wierenga  _et al._ [2018] H. Wierenga, P. R. ten Wolde, and N. B. Becker, Quantifying fluctuations in reversible enzymatic cycles and clocks, Phys. Rev. E 97, 042404 (2018).
  * Mora and Wingreen [2010] T. Mora and N. S. Wingreen, Limits of Sensing Temporal Concentration Changes by Single Cells, Phys. Rev. Lett. 104, 248101 (2010).
  * Nagase and Sagawa [2024] R. Nagase and T. Sagawa, Thermodynamically Optimal Information Gain in Finite Time Measurement (2024), arxiv:2307.16757 [cond-mat] .
  * García-Pintos  _et al._ [2022] L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical speed limits on observables, Phys. Rev. X 12, 011038 (2022).
  * Zheng and Lu [2024] J. Zheng and Z. Lu, Universal fluctuation-response relations of nonequilibrium dynamics: A trajectory information geometry framework (2024), arXiv:2403.10952 [cond-mat] .
  * Skinner and Dunkel [2021] D. J. Skinner and J. Dunkel, Estimating entropy production from waiting time distributions, Phys. Rev. Lett. 127, 198101 (2021).
  * van der Meer  _et al._ [2022] J. van der Meer, B. Ertel, and U. Seifert, Thermodynamic inference in partially accessible Markov networks: A unifying perspective from transition-based waiting time distributions, Phys. Rev. X 12, 031025 (2022).
  * Harunari  _et al._ [2022] P. E. Harunari, A. Dutta, M. Polettini, and É. Roldán, What to Learn from a Few Visible Transitions’ Statistics?, Phys. Rev. X 12, 041026 (2022).
  * Pietzonka  _et al._ [2016] P. Pietzonka, A. C. Barato, and U. Seifert, Universal bounds on current fluctuations, Phys. Rev. E 93, 052145 (2016).
  * Maes [2017] C. Maes, Frenetic bounds on the entropy production, Phys. Rev. Lett. 119, 160601 (2017).
  * Shiraishi  _et al._ [2018] N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  * Vo  _et al._ [2020] V. T. Vo, T. Van Vu, and Y. Hasegawa, Unified approach to classical speed limit and thermodynamic uncertainty relation, Phys. Rev. E 102, 062132 (2020).
  * Van Vu and Saito [2023] T. Van Vu and K. Saito, Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits, Phys. Rev. X 13, 011013 (2023).
  * Stray  _et al._ [2022] B. Stray, A. Lamb, A. Kaushik, J. Vovrosh, A. Rodgers, J. Winch, F. Hayati, D. Boddice, A. Stabrawa, A. Niggebaum, M. Langlois, Y.-H. Lien, S. Lellouch, S. Roshanmanesh, K. Ridley, G. de Villiers, G. Brown, T. Cross, G. Tuckwell, A. Faramarzi, N. Metje, K. Bongs, and M. Holynski, Quantum sensing for gravity cartography, Nature 602, 590 (2022).
  * Rosenband  _et al._ [2008] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place, Science 319, 1808 (2008).
  * Derevianko and Pospelov [2014] A. Derevianko and M. Pospelov, Hunting for topological dark matter with atomic clocks, Nat. Phys. 10, 933 (2014).
  * Roussy  _et al._ [2023] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, A. Wang, J. Ye, and E. A. Cornell, An improved bound on the electron’s electric dipole moment, Science 381, 46 (2023).
  * Pietzonka [2022] P. Pietzonka, Classical pendulum clocks break the thermodynamic uncertainty relation, Phys. Rev. Lett. 128, 130606 (2022).
  * [100] Github repository: https://doi.org/10.5281/zenodo.11636733.
  * Cox [1982] D. R. Cox,  _Renewal Theory_ , repr ed., Monographs on Statistics and Applied Probability (Chapman & Hall, London, 1982).
  * David and Larry [1987] A. David and S. Larry, The least variable phase type distribution is Erlang, Commun. Stat. Stoch. Models 3, 467 (1987).
  * Knuth [1976] D. E. Knuth, Big omicron and big omega and big theta, ACM SIGACT News 8, 18 (1976).