  Self-testing is a device-independent method that usually amounts to show that
the maximal quantum violation of a Bell's inequality certifies a unique quantum
state, up to some symmetries inherent to the device-independent framework. In
this work, we enlarge this approach and show how a coarse-grained version of
self-testing is possible in which physically relevant properties of a many-body
system are certified. For that we study a Bell scenario consisting of an
arbitrary number of parties and show that the membership to a set of
(entangled) quantum states whose size grows exponentially with the number of
particles can be self-tested. Specifically, we prove that a many-body
generalization of the chained Bell inequality is maximally violated if and only
if the underlying quantum state is equal, up to local isometries, to a
many-body singlet. The maximal violation of the inequality therefore certifies
any statistical mixture of the exponentially-many orthogonal pures states
spanning the singlet manifold.
