  This work concerns the statistics of the Two-Time Measurement definition of
heat variation in each reservoir of a thermodynamic quantum system. We study
the cumulant generating function of the heat flows in the thermodynamic and
large-time limits. It is well-known that, if the system is time-reversal
invariant, this cumulant generating function satisfies the celebrated
Evans--Searles symmetry. We show in addition that, under appropriate
ultraviolet regularity assumptions on the local interaction between the
reservoirs, it satisfies a translation-invariance property, as proposed in
[Andrieux et al. New J. Phys. 2009]. We particularly fix some proofs of the
latter article where the ultraviolet condition was not mentioned. We detail how
these two symmetries lead respectively to fluctuation relations and a
statistical refinement of heat conservation for isolated thermodynamic quantum
systems. As in [Andrieux \emph{et al.} New J. Phys. 2009], we recover the
Fluctuation-Dissipation Theorem in the linear response theory, short of
Green--Kubo relations. We illustrate the general theory on a number of
canonical models.
