  We consider the temporal correlations of the quantum state of a qubit subject
to simultaneous continuous measurement of two non-commuting qubit observables.
Such qubit state correlators are defined for an ensemble of qubit trajectories,
which has the same fixed initial state and can also be optionally constrained
by a fixed final state. We develop a stochastic path integral description for
the continuous quantum measurement and use it to calculate the considered
correlators. Exact analytic results are possible in the case of ideal
measurements of equal strength and are also shown to agree with solutions
obtained using the Fokker-Planck equation. For a more general case with
decoherence effects and inefficiency, we use a diagrammatic approach to find
the correlators perturbatively in the quantum efficiency. We also calculate the
state correlators for the quantum trajectories which are extracted from readout
signals measured in a transmon qubit experiment, by means of the quantum
Bayesian state update. We find an excellent agreement between the correlators
based on the experimental data and those obtained from our analytical and
numerical results.
