# Memory cost of temporal correlations

## TL;DR

This paper investigates the memory cost of simulating temporal correlations in different theories, including classical, quantum, and general probability theories, revealing inequalities that distinguish these correlations.

## Contribution

It introduces a theory-independent framework to analyze the memory cost of temporal correlations, extending beyond quantum mechanics to general probability theories.

## Key findings

- Derived inequalities to distinguish classical, quantum, and GPT temporal correlations.
- Showed that systems with finite memory capacity can exhibit different temporal correlations.
- Provided a unified approach to compare nonclassicality across theories.

## Abstract

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.