  We consider Hamiltonian systems which can be described both classically and
quantum mechanically. Trace formulas establish links between the energy spectra
of the quantum description and the spectrum of actions of periodic orbits in
the classical description. This duality is investigated in the present paper.
The duality holds for chaotic as well as for integrable systems. For billiards
the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical
spectrum (lengths of periodic orbits) are two manifestations of the billiard's
boundary. The trace formula expresses this link as a Fourier transform relation
between the corresponding spectral densities. It follows that the two-point
statistics are also simply related. The universal correlations of the quantal
spectrum are well known, consequently one can deduce the classical universal
correlations. An explicit expression for the scale of the classical
correlations is derived and interpreted. This allows a further extension of the
formalism to the case of complex billiard systems, and in particular to the
most interesting case of diffusive system. The concept of classical
correlations allows a better understanding of the so-called diagonal
approximation and its breakdown. It also paves the way towards a semiclassical
theory that is capable of global description of spectral statistics beyond the
breaktime. An illustrative application is the derivation of the
disorder-limited breaktime in case of a disordered chain, thus obtaining a
semiclassical theory for localization. A numerical study of classical
correlations in the case of the 3D Sinai billiard is presented. We gain a
direct understanding of specific statistical properties of the classical
spectrum, as well as their semiclassical manifestation in the quantal spectrum.
