  In this paper, we analyze the evolution of the generalized Pauli channels
governed by the memory kernel master equation. We provide necessary and
sufficient conditions for the memory kernel to give rise to the legitimate
(completely positive and trace-preserving) quantum evolution. In particular, we
analyze a class of kernels generating the quantum semi-Markov evolution, which
is a natural generalization of the Markovian semigroup. Interestingly, the
convex combination of Markovian semigroups goes beyond the semi-Markov case.
Our analysis is illustrated with several examples.
