  We study the $n$-level spectral correlation functions of classically chaotic
quantum systems without time-reversal symmetry. According to Bohigas, Giannoni
and Schmit's universality conjecture, it is expected that the correlation
functions are in agreement with the prediction of the Circular Unitary Ensemble
(CUE) of random matrices. A semiclassical resummation formalism allows us to
express the correlation functions as sums over pseudo-orbits. Using an extended
version of the diagonal approximation on the pseudo-orbit sums, we derive the
$n$-level correlation functions identical to the $n \times n$ determinantal
correlation functions of the CUE.
