  The growth in the demand for precisely crafted many-body systems of
spin-$1/2$ particles/qubits is due to their top-notch versatility in
application-oriented quantum-enhanced protocols and the fundamental tests of
quantum theory. Here we address the question: how quantum is a chain of spins?
We demonstrate that a single element of the density matrix carries the answer.
Properly analyzed it brings information about the extent of the many-body
entanglement and the non-locality. This method can be used to tailor and
witness highly non-classical effects in many-body systems with possible
applications to quantum computing, ultra-precise metrology or large-scale tests
of quantum mechanics. As a proof of principle, we investigate the extend of
non-locality and entanglement in ground states and thermal states of
experimentally accessible spin chains.
