  Quantum non-locality is an important concept in quantum physics. In this
work, we study the quantum non-locality in a fermion many-body system under
quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is
systematically investigated, which quantifies quantum non-locality between two
sites. We find that the quantum non-locality explicitly characterize the
extended and critical phase transitions, and further that in the globally
averaged picture of maximum value of the quantum non-locality the CHSH
inequality is not broken, but for a local pair in the internal of the system
the violation probability of the CHSH inequality becomes sufficiently finite.
Further we investigate an extension of the CHSH inequality,
Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize
multipartite quantum non-locality. We also find a similar behavior to the case
of CHSH inequality. In particular, in the critical regime and on a transition
point, the adjacent three qubit MKS polynomial in a portion of the system
exhibits a quantum non-local violation regime with a finite probability.
