  The discrepancy between maximally entangled states and maximally
non-classical quantum correlations is well-known but still not well understood.
We aim to investigate the relation between quantum correlations and
entanglement in a family Bell inequalities with $N$-settings and $d$ outcomes.
Using analytical as well as numerical techniques, we derive both maximal
quantum violations and violations obtained from maximally entangled states.
Furthermore, we study the most non-classical quantum states in terms of their
entanglement entropy for large values of $d$ and many measurement settings.
Interestingly, we find that the entanglement entropy behaves very differently
depending on whether $N=2$ or $N> 2$: when $N=2$ the entanglement entropy is a
monotone function of $d$ and the most non-classical state is far from maximally
entangled, whereas when $N> 2$ the entanglement entropy is a non-monotone
function of $d$ and converges to that of the maximally entangled state in the
limit of large $d$.
