  The time evolution in quantum many-body systems after external excitations is
attracting high interest in many fields. The theoretical modeling of these
processes is challenging, and the only rigorous quantum-dynamics approach that
can treat correlated fermions in two and three dimensions is nonequilibrium
Green functions (NEGF). However, NEGF simulations are computationally expensive
due to their $T^3$-scaling with the simulation duration $T$. Recently,
$T^2$-scaling was achieved with the generalized Kadanoff--Baym ansatz (GKBA),
for the second-order Born (SOA) selfenergy, which has substantially extended
the scope of NEGF simulations. In a recent Letter [Schl\"unzen \textit{et al.},
Phys. Rev. Lett. \textbf{124}, 076601 (2020)] we demonstrated that GKBA-NEGF
simulations can be efficiently mapped onto coupled time-local equations for the
single-particle and two-particle Green functions on the time diagonal, hence
the method has been called G1--G2 scheme. This allows one to perform the same
simulations with order $T^1$-scaling, both for SOA and $GW$ selfenergies giving
rise to a dramatic speedup. Here we present more details on the G1--G2 scheme,
including derivations of the basic equations including results for a general
basis, for Hubbard systems and for jellium. Also, we demonstrate how to
incorporate initial correlations into the G1--G2 scheme. Further, the
derivations are extended to a broader class of selfenergies, including the $T$
matrix in the particle--particle and particle--hole channels, and the
dynamically screened-ladder approximation. Finally, we demonstrate that, for
all selfenergies, the CPU time scaling of the G1--G2 scheme with the basis
dimension, $N_b$, can be improved compared to our first report: the overhead
compared to the original GKBA, is not more than an additional factor $N_b$.
