Let $S$ be the set of all Boolean functions $f:\left\{  0,1\right\}  ^{n}\rightarrow\left\{
0,1\right\}  $ that are expressible as $\widehat{f}_{\xi }$ for some such advice state $\xi
$.  Then $S$ includes the ‘target function’ $f^{\ast}:=L_{n}$ (the restriction of $L$ to inputs of length $n$), as well as a potentially-large number of other functions.  However, we claim $S$ is not *too* large: $\left\vert S\right\vert \leq2^{\operatorname*{poly}\left(  n\right)  }$.  This bound on the ‘effective information content’ of quantum states was derived previously by Aaronson [@aar:adv; @aar:learn], building on the work of Ambainis et al. [@antv].

One might initially hope that, just by virtue of the size bound on $S$, we could find some set of $\operatorname*{poly}(n)$ values$$\left(  x_{1},f^{\ast}\left(  x_{1}\right)  \right)  ,\ldots,\left(
x_{k},f^{\ast}\left(  x_{t}\right)  \right)$$ which *isolate* $f^{\ast}$ in $S$—that is, which differentiate $f^{\ast}$ from all other members of $S$.  In that case, the trusted classical advice could simply specify those values, as ‘tests’ for Arthur to perform on the quantum state sent by Merlin.  Alas, this hope is unfounded in general.  For consider the case where $f^{\ast}$ is the identically-zero function, and $S$ consists of $f^{\ast}$ along with the ‘point function’ $f_{y}$ (which equals $1$ on $y$ and $0$ elsewhere), for all $y\in\left\{
0,1\right\}  