  In the dynamics of open quantum systems, information may propagate in time
through either the system or the environment, giving rise to Markovian and
non-Markovian temporal correlations, respectively. However, despite their
notable coexistence in most physical situations, it is not yet clear how these
two quantities may limit the existence of one another. Here, we address this
issue by deriving several inequalities relating the temporal correlations of
general multi-time quantum processes. The dynamics are described by process
tensors and the correlations are quantified by the mutual information between
subsystems of their Choi states. First, we prove a set of upper bounds to the
non-Markovianity of a process given the degree of Markovianity in each of its
steps. This immediately implies a non-trivial maximum value for the
non-Markovianity of any process, independently of its Markovianity. Finally, we
obtain how the non-Markovianity limits the amount of total temporal
correlations that could be present in a given process. These results show that,
although any multi-time process must pay a price in total correlations to have
a given amount of non-Markovianity, this price vanishes exponentially with the
number of steps of the process, while the maximum non-Markovianity grows only
linearly. This implies that even a highly non-Markovian process might be
arbitrarily close to having maximum total correlations if it has a sufficiently
large number of steps.
