  Preparation and measurement of physical systems are the operational building
blocks of any physical experiment, and to describe them is the first purpose of
any physical theory. It is remarkable that, in some situations, even when only
preparation and measurement devices of a single system are present and they are
uncharacterized, it is possible to distinguish between the behaviours of
quantum and classical systems relying only on observational data. Certifying
the physical origin of measurement statistics in the prepare and measure
scenario is of primal importance for developing quantum networks, distributing
quantum keys and certifying randomness, to mention a few applications, but,
surprisingly, no general methods to do so are known. We progress on this
problem by crafting a general, sufficient condition to certify that a given set
of preparations can only generate classical statistics, for any number of
generalized measurements. As an application, we employ the method to
demonstrate non-classicality activation in the prepare and measure scenario,
also considering its application in random access codes. Following that, we
adapt our method to certify, again through a sufficient condition, whether a
given set of measurements can never give rise to non-classical behaviors,
irrespective of what preparations they may act upon. This, in turn, allows us
to find a large set of incompatible measurements that cannot be used to
demonstrate non-classicality, thus showing incompatibility is not sufficient
for non-classicality in the prepare and measure scenario.
