Is Quadratically Constrained Quadratic Program (QCQP) in NP?

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The general version of QCQP is NP-hard, but is it also NP-complete? That means, is there a non-deterministic algorithm, which solves QCQP in polynomial time complexity?

If the general version of QCQP is not in NP, are there restricted versions such that the problem is in NP?