  In this paper, building on some recent progress combined with numerical
techniques, we shed some new light on how the nonlocality of symmetric states
is related to their entanglement properties and potential usefulness in quantum
information processing. We use semidefinite programming techniques to devise a
device independent classification of three four qubit states into two classes
inequivalent under local unitaries and permutation of systems (LUP). We study
nonlocal properties when the number of parties grows large for two important
classes of symmetric states: the W states and the GHZ states, showing that they
behave differently under the inequalities we consider. We also discuss the
monogamy arising from the nonlocal correlations of symmetric states. We show
that although monogamy in a strict sense is not guaranteed for all symmetric
states, strict monogamy is achievable for all Dicke states when the number of
parties goes to infinity, as shown by an inequality based on a recent work
studying the nonlocality (and showing strict monogamy) of W states.
