Identifying non-classicality unambiguously and inexpensively is a
long-standing open challenge in physics. The No-Signalling-In-Time protocol was
developed as an experimental test for macroscopic realism, and serves as a
witness of quantum coherence in isolated quantum systems by comparing the
quantum state to its completely dephased counterpart. We show that it provides
a lower bound on a certain resource-theoretic coherence monotone. We go on to
generalise the protocol to the case where the system of interest is coupled to
an environment. Depending on the manner of the generalisation, the resulting
witness either reports on system coherence alone, or on a disjunction of system
coherence with either (i) the existence of non-classical system-environment
correlations or (ii) non-negligible dynamics in the environment. These are
distinct failure modes of the Born approximation in non-isolated systems.Comment: 16pp, 2 figs, 5 thms. v2: typos corrected, references added and small
change to title to reflect that of published versio
