  We present an adaptation of the NPA hierarchy to the setting of synchronous
correlation matrices. Our adaptation improves upon the original NPA hierarchy
by using smaller certificates and fewer constraints, although it can only be
applied to certify synchronous correlations. We recover characterizations for
the sets of synchronous quantum commuting and synchronous quantum correlations.
For applications, we show that the existence of symmetric informationally
complete positive operator-valued measures and maximal sets of mutually
unbiased bases can be verified or invalidated with only two certificates of our
adapted NPA hierarchy.
