  Markovianity lies at the heart of communication problems. This in turn makes
the information-theoretic characterization of Markov processes worthwhile. Data
processing inequalities are ubiquitous in this sense, assigning necessary
conditions for all Markov processes. We address here the problem of the
information-theoretic analysis of constraints on Markov processes in the
quantum regime. We show the existence of a novel class of quantum data
processing inequalities called here quantum Markov monogamy inequalities. This
new class of necessary conditions on quantum Markov processes is inspired by
its counterpart for classical Markov processes, and thus providing a strong
link between classical and quantum constraints on Markovianity. We go on to
construct a family of multitime quantum Markov monogamy inequalities, based on
the process tensor formalism and that exploits multitime correlations. We then
show, by means of an explicit example, that the Markov monogamy inequalities
can be stronger than the usual quantum data processing inequalities.
