  Recent studies have revealed a deep connection between the asymmetry of
cross-correlations and thermodynamic quantities in the short-time limit. In
this study, we address the finite-time domain of the asymmetry for both open
classical and quantum systems. Focusing on Markovian dynamics, we show that the
asymmetry observed in finite-time cross-correlations is upper bounded by
dissipation. We prove that, for classical systems in a steady state with
arbitrary operational durations, the asymmetry exhibits, at most, linear growth
over time, with the growth speed determined by the rates of entropy production
and dynamical activity. In the long-time regime, the asymmetry exhibits
exponential decay, with the decay rate determined by the spectral gap of the
transition matrix. Remarkably, for quantum cases, quantum coherence is equally
important as dissipation in constraining the asymmetry of correlations. We
demonstrate an example where only quantum coherence bounds the asymmetry while
the entropy production rate vanishes. Furthermore, we generalize the short-time
bounds on correlation asymmetry, as reported by Shiraishi [Phys. Rev. E 108,
L042103 (2023)] and Ohga et al. [Phys. Rev. Lett. 131, 077101 (2023)], to
encompass finite-time scenarios. These findings offer novel insights into the
thermodynamic aspects of correlation asymmetry.
