  Correlated quantum many-particle systems out of equilibrium are of high
interest in many fields, including correlated solids, ultracold atoms or dense
plasmas. Accurate theoretical description of these systems is challenging both,
conceptionally and with respect to computational resources. We have recently
presented a quantum fluctuations approach which is equivalent to the
nonequilibrium $GW$ approximation [E. Schroedter \textit{et al.}, Cond. Matt.
Phys. \textbf{25}, 23401 (2022)] that promises high accuracy at low
computational cost. In a second publication [E. Schroedter \textit{et al.},
Phys. Rev. B \textbf{108}, 205109 (2023)], this approach was extended to the
two-time exchange-correlation functions and the density response properties.
Here, we analyze the properties of this approach in more detail. We demonstrate
that the method is equivalent to the Bethe--Salpeter equation for the two-time
exchange-correlation function when the generalized Kadanoff-Baym ansatz with
Hartree-Fock propagators is applied.
