  Bipartite and multipartite entangled states are basic ingredients for
constructing quantum networks and their accurate verification is crucial to the
functioning of the networks, especially for untrusted networks. Here we propose
a simple approach for verifying the Bell state in an untrusted quantum network
in which one party is not honest. Only local projective measurements are
required for the honest party. It turns out each verification protocol is tied
to a probability distribution on the Bloch sphere and its performance has an
intuitive geometric meaning. This geometric picture enables us to construct the
optimal and simplest verification protocols, which are also very useful to
detecting entanglement in the untrusted network. Moreover, we show that our
verification protocols can achieve almost the same sample efficiencies as
protocols tailored to standard quantum state verification. Furthermore, we
establish an intimate connection between the verification of
Greenberger-Horne-Zeilinger states and the verification of the Bell state. By
virtue of this connection we construct the optimal protocol for verifying
Greenberger-Horne-Zeilinger states and for detecting genuine multipartite
entanglement.
