  The analysis of a continuous measurement record $z(t)$ poses a fundamental
challenge in quantum measurement theory. Different approaches have been used in
the past as records can, e.g., exhibit predominantly Gaussian noise, telegraph
noise, or clicks at random times. The last case may appear as photon clicks in
an optical spin noise measurement at very low probe laser power. Here we show
that such random-time quantum measurements can similarly to the first two cases
be analyzed in terms of higher-order temporal correlations of the detector
output $z(t)$ and be related to the Liouvillian of the measured quantum system.
Our analysis in terms of up to fourth-order spectra (quantum polyspectra) shows
that this new type of spectra reveals the same valuable information as
previously studied higher-order spectra in case of usual continuous quantum
measurements. Surprisingly, broad-band system dynamics is revealed even for
deliberately low average measurement rates. Many applications are envisioned in
high-resolution spectroscopy, single-photon microscopy, circuit quantum
electrodynamics, quantum sensing, and quantum measurements in general.
