  We assess non-Markovianity of a quantum open-system dynamics through the
violation of temporal bell-like inequalities in a controllable Nuclear Magnetic
Resonance system. We investigate experimentally the connections between the
violation of the addressed temporal Bell-like inequality and the
non-divisibility of the effective evolution of our system, which we fully
characterize in a broad range of experimentally controllable regimes. Finally,
we investigate the link between our approach and Leggett-Garg-like inequalities
based on time-translational two-time correlation functions. These results open
up interesting perspectives for the effective, non-tomographic characterization
of dynamical evolution by combining tools of different nature.
