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We review and generalize the recently introduced framework of entropy vectorsfor detecting and quantifying genuine multipartite entanglement in highdimensional multicomponent quantum systems. We show that these ideas can beextended to discriminate among other forms of multipartite entanglement. Inparticular, we develop methods to test whether density matrices are:decomposable, i.\ e.\ separable with respect to certain given partitions of thesubsystems; $k$-separable, i.\ e.\ separable with respect to $k$ partitions ofthe subsystems; $k$-partite entangled, i.e. there is entanglement among asubset of at least $k$ parties. We also discuss how to asses the dimensionalityof entanglement in all these cases.

The entropy vector formalism and the structure of multidimensional entanglement in multipartite systems