Temporal witnesses of non-classicality and conservation laws

A general entanglement-based witness of non-classicality has recently been proposed, which can be applied to testing quantum effects in gravity. This witness is based on generating entanglement between two quantum probes via a mediator. In this paper we provide a "temporal" variant of this witness, using a single quantum probe to assess the non-classicality of the mediator. Within the formalism of quantum theory, we show that if a system $M$ is capable of inducing a coherent dynamical evolution of a quantum system $Q$, in the presence of a conservation law, then $M$ must be non-classical. This argument supports witnesses of non-classicality relying on a single quantum probe, which can be applied to a number of open issues, notably in quantum gravity or quantum biology.