  We study heat transfers in a single level quantum dot strongly coupled to
fermionic reservoirs and subjected to a time-dependent protocol modulating the
dot energy as well as the dot-reservoir coupling strength. The dynamics is
described using nonequilibrium Greens functions (NEGFs) evaluated to first
order beyond quasi-static driving. We show that any heat definition expressed
as an energy change in the reservoir energy plus any fraction of the
system-reservoir interaction is not an exact differential when evaluated along
reversible isothermal transformations, except when that fraction is zero.
However, even in that latter case the reversible heat divided by temperature,
namely the entropy, does not satisfy the third law of thermodynamics and
diverges in the low temperature limit. Our results cast doubts on the
possibility to define a thermodynamically consistent notion of heat expressed
as the expectation value of some Hamiltonian terms.
