  From a quantum information perspective, verifying quantum coherence in a
quantum experiment typically requires adjusting measurement settings or
changing inputs. A paradigmatic example is that of a double-slit experiment,
where observing the interference pattern on the screen in a series of
experimental settings where one, the other, and both slits are open
unambiguously proves quantum coherence. Here we show that this is not necessary
by verifying quantum coherence in a network scenario without the need for
inputs. We show that there exist probability distributions for joint outcomes
of three parties in a triangular network with independent sources that cannot
be replicated using classical resources. Furthermore, we generalize our results
to $n$-party networks and show that the discrepancy between correlations in
classical and quantum networks increases with the number of parties. To this
end, we derive nonlinear inequalities that are satisfied by classical
correlations and find quantum states that violate them.
