  There are multipartite entangled states in many-body systems which may be
potential resources in various quantum applications. There are lots of methods
to witness specific entangled systems. However, no efficient method is
available to explore many-body systems without multipartite entanglement. It
may provide necessary restrictions for experimental preparations of
multipartite entanglement. Our goal is to solve this problem for spin systems.
The maximal effective velocity with propagation of information is bounded in
quantum spin systems with short-range interactions from Lieb-Robinson's
inequalities. This implies two clustering theorems for ground states and
thermal states. With these propagation relations, we show that both the gapped
ground state and thermal state at an upper-bounded inverse temperature have no
genuine multipartite nonlocality when disjoint regions are far away from each
other. The present $n$-particle system shows only biseparable quantum
correlations when the propagation relations show exponential decay. Similar
result holds for spin systems with product states as initial states. These
results show interesting features of quantum many-body systems with exponential
decay of correlations.
