Quantum clocks and thermodynamics

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Recent work in quantum thermodynamics has shown a quantitative connection between our ability to measure time precisely via a physical clock, and the thermodynamic arrow of time (in terms of increasing entropy). It would be interesting to explore this connection further, going beyond the particular model studied initially, as well as to investigate the broader implications of using explicit quantum clocks to measure time (such as the effective dynamics relative to the clock-time). Two additional related questions are: How are equilibration and thermalisation affected if time is discrete, such as in a quantum cellular automata (a local translationally invariant model of quantum computation)? Could we use a quantum clock to create a superposition of two different time-orderings of measurements, such that the results could not be explained by a classical mixture of time-orderings?