This paper shows that the statistical correlations of observables can tell quantum time evolution apart from classical time evolution in closed systems. The authors generalize a previous idea called the "signals of the quantum universe" (SQU) and argue that quantum vacuum fluctuations produce a characteristic, time-independent pattern of correlations. Classical Hamiltonian systems that reproduce the same short‑range statistics must be in physically excited, time‑evolving states. Those classical correlations display poles at physical frequencies and evolve in time before decaying by a process called dephasing.

The researchers set up a test that compares lists of observables in finite‑dimensional theories, treated either as quantum mechanical or classical. They focus on closed systems that evolve under a Hamiltonian, the rule that gives the system its energy and dynamics. In the quantum case they study vacuum fluctuations — the random outcomes you get even when a system sits in its lowest energy state. In the classical case they consider ensembles of time‑dependent states with random amplitudes and phases.

At a high level, the distinction hinges on whether the observables commute with the Hamiltonian. If they commute, diagonalization makes quantum and classical probabilities look the same and no test is possible. But when the observables do not commute with the Hamiltonian, the quantum vacuum can show nontrivial fluctuations while remaining stationary. Those quantum correlators are controlled by energy gaps between the ground state and excited states and do not depend on time. Classical correlators, by contrast, contain terms set by differences of physical frequencies. If those frequencies line up (a resonance), correlations grow on short timescales but then fade at late times because the different phases wash out — that is dephasing.