  A possible notion of nonclassicality for single systems can be defined on the
basis of the notion of memory cost of classically simulating probabilities
observed in a temporal sequence of measurements. We further explore this idea
in a theory-independent framework, namely, from the perspective of general
probability theories (GPTs), which includes classical and quantum theory as
special examples. Under the assumption that each system has a finite memory
capacity, identified with the maximal number of states perfectly
distinguishable with a single measurement, we investigate what are the temporal
correlations achievable with different theories, namely, classical, quantum,
and GPTs beyond quantum mechanics. Already for the simplest nontrivial
scenario, we derive inequalities able to distinguish temporal correlations
where the underlying system is classical, quantum, or more general.
