  We present definitions of both Connes spectrum and strong Connes spectrum for
actions of compact quantum groups on C*-algebras and obtain necessary and
sufficient conditions for a crossed product to be a prime or a simple
C*-algebra. Our results extend to the case of compact quantum actions the
results in [8] which in turn, generalize results by Connes, Olesen and Pedersen
and Kishimoto for abelian group actions. We prove in addition that the Connes
spectra are closed under tensor products. These results are new for compact
nonabelian groups as well.
