  Two widely used but distinct approaches to the dynamics of open quantum
systems are the Nakajima-Zwanzig and time-convolutionless quantum master
equation, respectively. Although both describe identical quantum evolutions
with strong memory effects, the first uses a time-nonlocal memory kernel
$\mathcal{K}$, whereas the second achieves the same using a time-local
generator $\mathcal{G}$. Here we show that the two are connected by a simple
yet general fixed-point relation: $\mathcal{G} =
\hat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial
relations between the two completely different ways of computing the
time-evolution and combine their strengths. We first discuss the stationary
generator, which enables a Markov approximation that is both nonperturbative
and completely positive for a large class of evolutions. We show that this
generator is not equal to the low-frequency limit of the memory kernel, but
additionally "samples" it at nonzero characteristic frequencies. This clarifies
the subtle roles of frequency dependence and semigroup factorization in
existing Markov approximation strategies. Second, we prove that the fixed-point
equation sums up the time-domain gradient / Moyal expansion for the
time-nonlocal quantum master equation, providing nonperturbative insight into
the generation of memory effects. Finally, we show that the fixed-point
relation enables a direct iterative numerical computation of both the
stationary and the transient generator from a given memory kernel. For the
transient generator this produces non-semigroup approximations which are
constrained to be both initially and asymptotically accurate at each iteration
step.
