  Both conservation laws and practical restrictions impose symmetry constraints
on the dynamics of open quantum systems. In the case of time-translation
symmetry, which arises naturally in many physically relevant scenarios, the
quantum coherence between energy eigenstates becomes a valuable resource for
quantum information processing. In this work we identify the minimum amount of
decoherence compatible with this symmetry for a given population dynamics. This
yields a generalisation to higher-dimensional systems of the relation $T_2 \leq
2 T_1$ for qubit decoherence and relaxation times. It also enables us to
witness and assess the role of non-Markovianity as a resource for coherence
preservation and transfer. Moreover, we discuss the relationship between
ergodicity and the ability of Markovian dynamics to indefinitely sustain a
superposition of different energy states. Finally, we establish a formal
connection between the resource-theoretic and the master equation approaches to
thermodynamics, with the former being a non-Markovian generalisation of the
latter. Our work thus brings the abstract study of quantum coherence as a
resource towards the realm of actual physical applications.
