Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, a generalized Spekkens non-contextual ontological model representation of the quantum statistics associated to the prepare-and-measure scenario is constructed. Any prepare-and-measure scenario is either classical or non-classical depending on whether it admits such a representation... A new mathematical criterion, called unit separability, is formulated as the relevant classicality criterion --- the name is inspired by the usual notion of quantum state separability. Using this criterion, we first derive simple upper and lower bounds on the cardinality of the ontic space. Then, we recast the unit separability criterion as a possibly infinite set of linear constraints to be verified, from which we derive two separate converging hierarchies of algorithmic tests to witness non-classicality or certify classicality. We relate the complexity of these algorithmic tests to that of a class of vertex enumeration problems. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories. (read more)