In classical systems, we often study two extreme regimes of motion: Regular integrable motion with many con- served quantities and irregular chaotic motion with exponential sensitivity to initial conditions. Characterizing chaos in quantum systems is more difficult. Studying quantum systems with a chaotic classical limit reveals the universal characteristics chaotic quantum systems usually have. It is conjectured that many of their statistical properties, such as the energy level spacing distribution, can be modeled with ensembles of random matrices, where the appropriate ensemble is determined only by the type of time-reversal symmetry of the system. This conjecture is numerically and experimentally well-tested. In this paper we demonstrate it on the example of quantum kicked tops. In contrast, the statistical properties of integrable quantum systems cannot be modelled by random matrices, making them special compared to the generic chaotic quantum systems.

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