Thermal Operations in general are not memoryless
So-called Thermal Operations seem to describe the most fundamental, and reasonable, set of operations allowable for state transformations at an ambient inverse temperature $\beta$. However, a priori, they require experimentalists to manipulate very complex environments and have control over their internal degrees of freedom. For this reason, the community has been working on creating more experimentally-friendly operations. In [Perry et al., Phys. Rev. X 8, 041049] it was shown that for states diagonal in the energy basis, that Thermal Operations can be performed by so-called Coarse Operations, which need just one auxiliary qubit, but are otherwise Markovian and classical in spirit. In this work, by providing an explicit counterexample, we show that this one qubit of memory is necessary. We also fully characterize the possible transitions that do not require memory for the system being a qubit. We do this by analyzing arbitrary control sequences comprising level energy changes and partial thermalizations in each step.