  Bipartite and multipartite entangled states are of central interest in
quantum information processing and foundational studies. Efficient verification
of these states, especially in the adversarial scenario, is a key to various
applications, including quantum computation, quantum simulation, and quantum
networks. However, little is known about this topic in the adversarial
scenario. Here we initiate a systematic study of pure-state verification in the
adversarial scenario. In particular, we introduce a general method for
determining the minimal number of tests required by a given strategy to achieve
a given precision. In the case of homogeneous strategies, we can even derive an
analytical formula. Furthermore, we propose a general recipe to verifying pure
quantum states in the adversarial scenario by virtue of protocols for the
nonadversarial scenario. Thanks to this recipe, the resource cost for verifying
an arbitrary pure state in the adversarial scenario is comparable to the
counterpart for the nonadversarial scenario, and the overhead is at most three
times for high-precision verification. Our recipe can readily be applied to
efficiently verify bipartite pure states, stabilizer states, hypergraph states,
weighted graph states, and Dicke states in the adversarial scenario, even if
only local projective measurements are accessible. This paper is an extended
version of the companion paper Zhu and Hayashi, Phys. Rev. Lett. 123, 260504
(2019).
