  Multipartite quantum states may exhibit different types of quantum
entanglement in that they cannot be converted into each other by local quantum
operations only, and fully understanding mathematical structures of different
types of multipartite entanglement is a very challenging task. In this paper,
from the viewpoint of Hardy's nonlocality, we compare W and GHZ states and show
a couple of crucial different behaviors between them. Particularly, by
developing a geometric model for the Hardy's nonlocality problem of W states,
we derive an upper bound for its maximal violation probability, which turns out
to be strictly smaller than the corresponding probability of GHZ state. This
gives us a new comparison between these two quantum states, and the result is
also consistent with our intuition that GHZ states is more entangled.
Furthermore, we generalize our approach to obtain an asymptotic
characterization for general $N$-qubit W states, revealing that when $N$ goes
up, the speed that the maximum violation probabilities decay is exponentially
slower than that of general $N$-qubit GHZ states. We provide some numerical
simulations to verify our theoretical results.
