  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.
