  We present a quantum version of the generalized $(h,\phi)$-entropies,
introduced by Salicr\'u \textit{et al.} for the study of classical probability
distributions. We establish their basic properties, and show that already known
quantum entropies such as von Neumann, and quantum versions of R\'enyi,
Tsallis, and unified entropies, constitute particular classes of the present
general quantum Salicr\'u form. We exhibit that majorization plays a key role
in explaining most of their common features. We give a characterization of the
quantum $(h,\phi)$-entropies under the action of quantum operations, and study
their properties for composite systems. We apply these generalized entropies to
the problem of detection of quantum entanglement, and introduce a discussion on
possible generalized conditional entropies as well.
