  There exist two canonical approaches to describe open quantum systems by a
time-evolution equation: the Nakajima-Zwanzig quantum master equation,
featuring a time-nonlocal memory kernel $\mathcal{K}$, and the
time-convolutionless equation with a time-local generator $\mathcal{G}$. These
key quantities have recently been shown to be connected by an exact fixed-point
relation [Phys. Rev. X 11, 021041 (2021)]. Here we show that this implies a
recursive relation between their perturbative expansions, allowing a series for
the kernel $\mathcal{K}$ to be translated directly into a corresponding series
for the more complicated generator $\mathcal{G}$. This leads to an elegant way
of computing the generator using well-developed, standard memory-kernel
techniques for strongly interacting open systems. Moreover, it allows for an
unbiased comparison of time-local and time-nonlocal approaches independent of
the particular technique chosen to calculate expansions of $\mathcal{K}$ and
$\mathcal{G}$ (Nakajima-Zwanzig projections, real-time diagrams, etc.). We
illustrate this for leading and next-to-leading order calculations of
$\mathcal{K}$ and $\mathcal{G}$ for the single impurity Anderson model using
both the bare expansion in the system-environment coupling and a more advanced
renormalized series. We compare the different expansions obtained, quantify the
legitimacy of the generated dynamics (complete positivity) and benchmark with
the exact result in the non-interacting limit.
