  Bell non-locality represents the ultimate consequence of quantum
entanglement, fundamentally undermining the classical tenet that
spatially-separated degrees of freedom possess objective attributes
independently of the act of their measurement. Despite its importance, probing
Bell non-locality in many-body systems is considered to be a formidable
challenge, with a computational cost scaling exponentially with system size.
Here we propose and validate an efficient variational scheme, based on the
solution of inverse classical Ising problems, which in polynomial time can
probe whether an arbitrary set of quantum data is compatible with a local
theory; and, if not, it delivers a many-body Bell inequality violated by the
quantum data. We use our approach to unveil new many-body Bell inequalities,
violated by suitable measurements on paradigmatic quantum states (the
low-energy states of Heisenberg antiferromagnets), paving the way to systematic
Bell tests in the many-body realm.
