  We consider the evolution of two-time correlations in the quantum XXZ
spin-chain in contact with an environment causing dephasing. Extending
quasi-exact time-dependent matrix product state techniques to consider the
dynamics of two-time correlations within dissipative systems, we uncover the
full quantum behavior for these correlations along all spin directions.
Together with insights from adiabatic elimination and kinetic Monte Carlo, we
identify three dynamical regimes. For initial times, their evolution is
dominated by the system unitary dynamics and depends on the initial state and
the Hamiltonian parameters. For weak spin-spin interaction anisotropy, after
this initial dynamical regime, two-time correlations enter an algebraic scaling
regime signaling the breakdown of time-translation invariance and the emergence
of aging. For stronger interaction anisotropy, these correlations first go
through a stretched exponential regime before entering the algebraic one. Such
complex relaxation arises due to the competition between the proliferation
dynamics of energetically costly excitations and their motion. As a result,
dissipative heating dynamics of spin systems can be used to probe the entire
spectrum of the underlying Hamiltonian.
