  Open quantum systems can exhibit complex states, which classification and
quantification is still not well resolved. The Kerr-nonlinear cavity,
periodically modulated in time by coherent pumping of the intra-cavity photonic
mode, is one of the examples. Unraveling the corresponding Markovian master
equation into an ensemble of quantum trajectories and employing the recently
proposed calculation of quantum Lyapunov exponents [I.I. Yusipov {\it et al.},
Chaos {\bf 29}, 063130 (2019)], we identify `chaotic' and `regular' regimes
there. In particular, we show that chaotic regimes manifest an intermediate
power-law asymptotics in the distribution of photon waiting times. This
distribution can be retrieved by monitoring photon emission with a
single-photon detector, so that chaotic and regular states can be discriminated
without disturbing the intra-cavity dynamics.
