  We introduce and study a class of entanglement criteria based on the idea of
applying local contractions to an input multipartite state, and then computing
the projective tensor norm of the output. More precisely, we apply to a mixed
quantum state a tensor product of contractions from the Schatten class $S_1$ to
the Euclidean space $\ell_2$, which we call entanglement testers. We analyze
the performance of this type of criteria on bipartite and multipartite systems,
for general pure and mixed quantum states, as well as on some important classes
of symmetric quantum states. We also show that previously studied entanglement
criteria, such as the realignment and the SIC POVM criteria, can be viewed
inside this framework. This allows us to answer in the positive two conjectures
of Shang, Asadian, Zhu, and G\"uhne by deriving systematic relations between
the performance of these two criteria.
