  We investigate both analytically and numerically violations of a Leggett-Garg
inequality (LGI) for a composite quantum system in contact with two separate
reservoirs at different temperatures. Remarkably we find that LGI violations
can be enhanced when a heat current is established at low temperatures in a
steady-state regime. Based on a Kraus operator decomposition of the non-unitary
evolution for a system formed by two interacting spins or quantum bits, we
provide analytical support for power law relations between dissipation strength
and mean temperature in the borderlines separating parameter regions where
non-equilibrium conditions affect differently the maximal LGI violation.
Furthermore, a correspondence between spatial and temporal correlation
inequalities is shown to persist even in such nonequilibrium thermal settings.
