the matrix elements are given by simple algebraic expressions because computing matrix elements can also be a computational burden. A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos. Correlating statistical descriptions of quantum mechanics with classical behavior Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Random matrix theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute. A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees of freedom (Berry and Tabor have put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al. present a semiclassical explanation of the so-called Bohigas–Giannoni–Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos. Qualitative observations of level repulsions can be quantified and related to the classical dynamics using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson distribution of energy levels: In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic