  Here we investigate the impact of temporal entanglement on a system's ability
to perform thermodynamical work. We show that while the quantum version of the
Jarzynski equality remains satisfied even in the presence of temporal
entanglement, the individual thermodynamical work moments in the expansion of
the free energy are, in fact, sensitive to the genuine quantum correlations.
Therefore, while individual moments of the amount of thermodynamical work can
be larger (or smaller) quantumly than classically, when they are all combined
together into the (exponential of) free energy, the total effect vanishes to
leave the Jarzynski equality intact. Whether this is a fortuitous coincidence
remains to be seen, but it certainly goes towards explaining why the laws of
thermodynamics happen to be so robust as to be independent of the underlying
micro-physics. We discuss the relationship between this result and
thermodynamical witnesses of spatial entanglement as well as explore the subtle
connection with the "quantum arrow of time".
