  The problem of closing the detection loophole in Bell tests is investigated
in the presence of a limited number of efficient detectors using emblematic
multipartite quantum states. To this end, a family of multipartite Bell
inequalities is introduced basing on local projective measurements conducted by
$N-k$ parties and applying a $k$-party Bell inequality on the remaining
parties. Surprisingly, we find that most of the studied pure multipartite
states involving e.g. cluster states, the Dicke states, and the
Greenberger-Horne-Zeilinger states can violate our inequalities with only the
use of two efficient detectors, whereas the remaining detectors may have
arbitrary small efficiencies. We believe that our inequalities are useful in
Bell experiments and device-independent applications if only a small number of
highly efficient detectors are in our disposal.
