  Non-classical probability is the underlying feature of quantum mechanics. The
emergence of Bell-CHSH non-locality for bipartite systems and linear
entanglement inequalities for two-qubit systems has been shown in Adhikary et
al. 2020 [Eur. Phys. J. D 74, 68 (2020)], purely as violations of classical
probability rules. In this paper, we improve upon that work by showing that
violation of any nonlocality inequality implies violation of classical
probability rules, manifested through negative probabilities, without recourse
to any underlying theory. Moving on to entanglement, we employ parent
pseudoprojections to show how any number of linear and nonlinear entanglement
witnesses for multiqubit systems can be obtained as violations of classical
probability rules. They include the ones that have been derived earlier by
employing different methods. It provides a perspective complementary to the
current understanding in terms of the algebraic approaches.
