  We investigate the role of nonclassical temporal correlations in enhancing
the performance of ticking clocks in a discrete-time scenario. We show that the
problem of optimal models for ticking clocks is related to the violation of
Leggett-Garg-type temporal inequalities formulated in terms of, possibly
invasive, sequential measurements, but on a system with a bounded memory
capacity. Ticking clocks inspire the derivation of a family of temporal
inequalities showing a gap between classical and quantum correlations, despite
involving no input. We show that quantum ticking-clock models achieving
accuracy beyond the classical bound are also those violating Leggett-Garg-type
temporal inequalities for finite sequences and we investigate their
continuous-time limit. Interestingly, we show that optimal classical clock
models in the discrete-time scenario do not have a well-defined continuous-time
limit, a feature that is absent in quantum models.
