Quantum coherence generated in a physical process can only be cast as a potentially useful resource if its effects can be detected at a later time. Recently, the notion of non-coherence-generating-and-detecting (NCGD) dynamics has been introduced and related to the classicality of the statistics associated with sequential measurements at different times. However, in order for a dynamics to be NCGD, its propagators need to satisfy a given set of conditions for $all$ triples of consecutive times. We reduce this to a finite set of $d(d-1)$ conditions, where $d$ is the dimension of the quantum system, provided that the generator is time-independent. Further conditions are derived for the more general time-dependent case. The application of this result to the case of a qubit dynamics allows us to elucidate which kind of noise gives rise to non-coherence-generation-and-detection.
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