quantum bound. The objective of this article is to explain in detail the methods used to derive these results, along with the other results that were summarized in Ref. \[\].

![\[Fig:summary\] The thick black curves are qualitative sketches of the maximum efficiency as a function of heat-engine power output (main plot), or refrigerator cooling power (inset), with the shaded regions being forbidden. Precise plot of such curves for different temperature ratios, $T_R/T_L$, are shown in Fig. \[Fig:allpowers\]. The colored loops (red, grey and blue) are typical sketches of the efficiency versus power of [*individual*]{} heat-engines as we increase the load resistance (direction of arrows on loop). The power output $P_{\rm gen}=IV$ vanishes when the load resistance is zero (for which $V=0$) or infinite (for which $I=0$), with a maximum at an intermediate resistance (open square). The curves have a characteristic loop form [@Casati-review], however the exact shape of the loop depends on many system specific details, such as charging effects. The dashed blue loop is for a typical non-optimal system (always well below the upper bound), while the solid red and grey loops are for systems which achieve the upper bound for a particular value of the load. The star 