  Methods derived from the generalized quantum master equation (GQME) framework
have provided the basis for elucidating energy and charge transfer in systems
ranging from molecular solids to photosynthetic complexes. Recently, the
non-perturbative combination of the GQME with quantum-classical methods has
resulted in approaches whose accuracy and efficiency exceed those of the
original quantum-classical schemes while offering significant accuracy
improvements over perturbative expansions of the GQME. Here we show that, while
the non-Markovian memory kernel required to propagate the GQME scales
quartically with the number of subsystem states, the number of trajectories
required scales at most quadratically when using quantum-classical methods to
construct the kernel. We then present an algorithm that allows further
acceleration of the quantum-classical GQME by providing a way to selectively
sample the kernel matrix elements that are most important to the process of
interest. We demonstrate the utility of these advances by applying the
combination of Ehrenfest mean field theory with the GQME (MF-GQME) to models of
the Fenna-Matthews-Olson (FMO) complex and the light harvesting complex II
(LHCII), with 7 and 14 states, respectively. This allows us to show that
MF-GQME is able to accurately capture all the relevant dynamical time-scales in
LHCII: the initial nonequilibrium population transfer on the femtosecond
time-scale, the steady state-type trapping on the picosecond time-scale, and
the long time population relaxation. Remarkably, all of these physical effects
spanning tens of picoseconds can be encoded in a memory kernel that decays
after only $\sim$65 fs.
