Distributed quantum information in networks is paramount for global secure
quantum communication. Moreover, it finds applications as a resource for
relevant tasks, such as clock synchronization, magnetic field sensing, and
blind quantum computation. For quantum network analysis and benchmarking of
implementations, however, it is crucial to characterize the topology of
networks in a way that reveals the nodes between which entanglement can be
reliably distributed. Here, we demonstrate an efficient scheme for this
topology certification. Our scheme allows for distinguishing, in a scalable
manner, different networks consisting of bipartite and multipartite
entanglement sources, for different levels of trust in the measurement devices
and network nodes. We experimentally demonstrate our approach by certifying the
topology of different six-qubit networks generated with polarized photons,
employing active feed-forward and time multiplexing. Our methods can be used
for general simultaneous tests of multiple hypotheses with few measurements,
being useful for other certification scenarios in quantum technologies.Comment: 18 pages, 5 figure