D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] constructed
N-qubit bound entangled states which violate a Bell inequality for $N\ensuremath{\ge}8$, and his result was recently improved by showing that there exists an
N-qubit bound entangled state violating the same Bell inequality if and only if $N\ensuremath{\ge}6$ [Phys. Rev. A 79, 032309 (2009)]. On the other hand, it has been also shown that the states which D\"ur considered violate Bell inequalities different from the inequality for $N\ensuremath{\ge}6$. In this paper, by employing different forms of Bell inequalities, in particular, a specific form of Bell inequalities with
M settings of the measuring apparatus for sufficiently large
M, we prove that there exists an
N-qubit bound entangled state violating the
M-setting Bell inequality if and only if $N\ensuremath{\ge}4$.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.