  The modeling of natural phenomena via a Markov process --- a process for
which the future is independent of the past, given the present--- is ubiquitous
in many fields of science. Within this context, it is of foremost importance to
develop ways to check from the available empirical data if the underlying
mechanism is indeed Markovian. A paradigmatic example is given by data
processing inequalities, the violation of which is an unambiguous proof of the
non-Markovianity of the process. Here, our aim is twofold. First we show the
existence of a monogamy-like type of constraints, beyond data processing,
respected by Markov chains. Second, to show a novel connection between the
quantification of causality and the violation of both data processing and
monogamy inequalities. Apart from its foundational relevance in the study of
stochastic processes we also consider the applicability of our results in a
typical quantum information setup, showing it can be useful to witness the
non-Markovianity arising in a sequence of quantum non-projective measurements.
