  We study the verification of maximally entangled states by virtue of the
simplest measurement settings: local projective measurements without adaption.
We show that optimal protocols are in one-to-one correspondence with complex
projective 2-designs constructed from orthonormal bases. Optimal protocols with
minimal measurement settings are in one-to-one correspondence with complete
sets of mutually unbiased bases. Based on this observation, optimal protocols
are constructed explicitly for any local dimension, which can also be applied
to estimating the fidelity with the target state and to detecting entanglement.
In addition, we show that incomplete sets of mutually unbiased bases are
optimal for verifying maximally entangled states when the number of measurement
settings is restricted. Moreover, we construct optimal protocols for the
adversarial scenario in which state preparation is not trusted. The number of
tests has the same scaling behavior as the counterpart for the nonadversarial
scenario; the overhead is no more than three times. We also show that the
entanglement of the maximally entangled state can be certified with any given
significance level using only one test as long as the local dimension is large
enough.
