  Armed with quantum correlations, quantum sensors in a network have shown the
potential to outclass their classical counterparts in distributed sensing tasks
such as clock synchronization and reference frame alignment. On the other hand,
this analysis was done for simple and idealized networks, whereas the
correlation shared within a practical quantum network, captured by the notion
of network states, is much more complex. Here, we prove a general bound that
limits the performance of using quantum network states to estimate a global
parameter, establishing the necessity of genuine multipartite entanglement for
achieving a quantum advantage. The bound can also serve as an entanglement
witness in networks and can be generalized to states generated by shallow
circuits. Moreover, while our bound prohibits local network states from
achieving the Heisenberg limit, we design a probabilistic protocol that, once
successful, attains this ultimate limit of quantum metrology and preserves the
privacy of involved parties. Our work establishes both the limitation and the
possibility of quantum metrology within quantum networks.
