  Statistical correlations that can be generated across the nodes in a quantum
network depend crucially on its topology. However, this topological information
might not be known a priori, or it may need to be verified. In this paper, we
propose an efficient protocol for distinguishing and inferring the topology of
a quantum network. We leverage entropic quantities -- namely, the von Neumann
entropy and the measured mutual information -- as well as measurement
covariance to uniquely characterize the topology. We show that the entropic
quantities are sufficient to distinguish two networks that prepare GHZ states.
Moreover, if qubit measurements are available, both entropic quantities and
covariance can be used to infer the network topology without state-preparation
assumptions. We show that the protocol can be entirely robust to noise and can
be implemented via quantum variational optimization. Numerical experiments on
both classical simulators and quantum hardware show that covariance is
generally more reliable for accurately and efficiently inferring the topology,
whereas entropy-based methods are often better at identifying the absence of
entanglement in the low-shot regime.
