  We present general quantum mechanical expressions for higher order moments,
cumulants, and spectra of continuously measured quantum systems with
applications in spin noise spectroscopy, quantum transport, and measurement
theory in general. Starting from the so-called stochastic master equation of
continuous measurement theory, we find that the leading orders of the
fluctuating detector output $z(t)$ with respect to the measurement strength
$\beta$ are a white shot noise background, a constant measurement offset, and
the leading order quantum noise of the measured operator $A$. Starting from
quantum expressions for the multi-time moments $\langle z(t_n)\cdots z(t_1)
\rangle$ we derive three- and four-time cumulants that are valid in all orders
of $\beta$ covering the full regime between the weak and strong measurement
limit (Zeno-limit). Intriguingly, quantum expressions for the cumulants were
found that exhibit the same simple structure as those for the moments after
introduction of only a slightly modified system propagator. Very compact
expressions for the cumulant-based third and fourth order spectra (bispectrum
and trispectrum) follow naturally. We illustrate the usefulness of higher order
spectra by treating a real world two-spin system with strong hyperfine
interaction.Moreover, spin noise spectroscopy is shown to have the potential
for investigating the transition from weak measurements to the famous quantum
Zeno regime for realistic probe laser intensities.
