  How can a multipartite single-photon path-entangled state be certified
efficiently by means of local measurements? We address this question by
constructing an entanglement witness based on local photon detections preceded
by displacement operations to reveal genuine multipartite entanglement. Our
witness is defined as a sum of three observables that can be measured locally
and assessed with two measurement settings for any number of parties $N$. For
any bipartition, the maximum mean value of the witness observable over
biseparable states is bounded by the maximum eigenvalue of an $N\times N$
matrix, which can be computed efficiently. We demonstrate the applicability of
our scheme by experimentally testing the witness for heralded 4- and 8-partite
single-photon path-entangled states. Our implementation shows the scalability
of our witness and opens the door for distributing photonic multipartite
entanglement in quantum networks at high rates.
