  A method is presented for the systematic derivation of a hierarchy of coupled
equations for the computation of two-time correlation functions of operators
for open many-body quantum systems. We show how these systems of equations can
be closed in mean-field and beyond approximations. Results for the specific
example of the spectral weight functions are discussed. Our method allows one
to access the full temporal evolution, not just the stationary solution, of
non-equilibrium open quantum problems described by a Markovian master equation.
