  The characterization of multi-time correlations in open quantum systems is of
fundamental importance. In this work, we investigate multi-time processes using
the process matrix formalism and show that the presence of a quantum
non-Markovian environment plays a significant role in enhancing the
communication capacity in sequential prepare-transform-measure Quantum Random
Access Codes (QRAC). The correlated environment enables a quantum advantage to
multiple parties, even with projective measurements. In particular, we show
that the Markovian and classical non-Markovian processes, i.e. quantum
processes with classical feedback from the environment, do not yield sequential
quantum advantage. In contrast, it is possible to achieve an advantage in the
presence of a quantum non-Markovian environment. Therefore this approach allows
a semi-device-independent certification of quantum non-Markovianity. As opposed
to entanglement-detection criteria which require the knowledge of the complete
process, this method allows to certify the presence of a quantum non-Markovian
environment from the observed measurement statistics. Moreover, quantum memory
ameliorates the unambiguous certifiable region of unsharp instruments in a
semi-device-independent manner.
