  The out-of-time-ordered correlator has been proposed as an indicator of chaos
in quantum systems due to its simple interpretation in the semiclassical limit.
In particular, its rate of possible exponential growth at $\hbar \to 0$ is
closely related to the classical Lyapunov exponent. Here we explore how this
approach to quantum chaos relates to the random-matrix theoretical description.
To do so, we introduce and study the level statistics of the logarithm of the
out-of-time-ordered operator, $\hat{\Lambda}(t) = \ln \left( -
\left[\hat{x}(t),\hat{p}_x(0) \right]^2 \right)/(2t)$, that we dub the
"Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level
statistics is calculated explicitly for the quantum stadium billiard. It is
shown that in the bulk of the filtered spectrum, this statistics perfectly
aligns with the Wigner-Dyson distribution. One of the advantages of looking at
the spectral statistics of this operator is that it has a well-defined
semiclassical limit where it reduces to the matrix of uncorrelated classical
finite-time Lyapunov exponents in a partitioned phase space. We provide a
heuristic picture interpolating these two limits using Moyal quantum mechanics.
Our results show that the Lyapunov operator may serve as a useful tool to
characterize quantum chaos and in particular quantum-to-classical
correspondence in chaotic systems, by connecting the semiclassical Lyapunov
growth at early times, when the quantum effects are weak, to universal level
repulsion that hinges on strong quantum interference effects.
