  The Clauser-Horne-Shimony-Holt inequality was originally proposed as a Bell
inequality to detect nonlocality in bipartite systems. However, it can also be
used to certify the nonlocality of multipartite quantum states. We apply this
to study the nonlocality of multipartite Greenberger-Horne-Zeilinger, W and
graph states under local decoherence processes. We derive lower bounds on the
critical local-noise strength tolerated by GHZ states before becoming local. In
addition, for the whole noisy dynamics, we derive lower bounds on the
corresponding nonlocal content for the three classes of states. All the bounds
presented can be calculated efficiently and, in some cases, provide
significantly tighter estimates than with any other known method. For example,
they reveal that $N$-qubit GHZ states undergoing local dephasing are, for all
$N$, nonlocal throughout all the dephasing dynamics.
