  Quantum entanglement and quantum entropy are crucial concepts in the study of
multipartite quantum systems. In this work we show how the notion of
concurrence vector, re-expressed in a particularly useful form, provides new
insights and computational tools for the analysis of both. In particular, using
this approach for a general multipartite pure state, one can easily prove known
relations in an easy way and to build up new relations between the concurrences
associated with the different bipartitions. The approach is also useful to
derive sufficient conditions for genuine entanglement in generic multipartite
systems that are computable in polynomial time. From an entropy-of-entanglement
perspective, the approach is powerful to prove properties of the Tsallis-$2$
entropy, such as the subadditivity, and to derive new ones, e.g. a modified
version of the strong subadditivity which is always fulfilled; thanks to the
purification theorem these results hold for any multipartite state, whether
pure or mixed.
