  Continuous-time Markovian evolution appears to be manifestly different in
classical and quantum worlds. We consider ensembles of random generators of
$N$-dimensional Markovian evolution, quantum and classical ones, and evaluate
their universal spectral properties. We then show how the two types of
generators can be related by superdecoherence. In analogy with the mechanism of
decoherence, which transforms a quantum state into a classical one,
superdecoherence can be used to transform a Lindblad operator (generator of
quantum evolution) into a Kolmogorov operator (generator of classical
evolution). We inspect spectra of random Lindblad operators undergoing
superdecoherence and demonstrate that, in the limit of complete
superdecoherence, the resulting operators exhibit spectral density typical to
random Kolmogorov operators. By gradually increasing strength of
superdecoherence, we observe a sharp quantum-to-classical transition.
Furthermore, we define an inverse procedure of supercoherification that is a
generalization of the scheme used to construct a quantum state out of a
classical one. Finally, we study microscopic correlation between neighbouring
eigenvalues through the complex spacing ratios and observe the horse-shoe
distribution, emblematic of the Ginibre universality class, for both types of
random generators. Remarkably, it survives superdecoherence and
supercoherification.
