  Generalized master equations provide a theoretically rigorous framework to
capture the dynamics of processes ranging from energy harvesting in plants and
photovoltaic devices, to qubit decoherence in quantum technologies, and even
protein folding. At their center is the concept of memory. The explicit
time-nonlocal description of memory is both protracted and elaborate. When
physical intuition is at a premium one would desire a more compact, yet
complete, description. Here, we demonstrate how and when the
time-convolutionless formalism constitutes such a description. In particular,
by focusing on the dissipative dynamics of the spin-boson and Frenkel exciton
models, we show how to: easily construct the time-local generator from
reference reduced dynamics, elucidate the dependence of its existence on the
system parameters and the choice of reduced observables, identify the physical
origin of its apparent divergences, and offer analysis tools to diagnose their
severity and circumvent their deleterious effects. We demonstrate that, when
applicable, the time-local approach requires as little information as the more
commonly used time-nonlocal scheme, with the important advantages of providing
a more compact description, greater algorithmic simplicity, and physical
interpretability. We conclude by introducing the discrete-time analogue and a
straightforward protocol to employ it in cases where the reference dynamics
have limited resolution. The insights we present here offer the potential for
extending the reach of dynamical methods, reducing both their cost and
conceptual complexity.
