Hyperdeterminants are high dimensional analogues of determinants, associated with tensors of formats generalizing square matrices. First conceived for $2\times 2\times 2$ tensors by Cayley, they were developed in generality by Gelfand, Kapranov and Zelevinsky. Yet, hyperdeterminants in three or more dimensions are long conjectured to be VNP-Hard to compute, akin to permanents and unlike determinants. Even deciding if the hyperdeterminant of a given tensor is zero is conjectured to be NP-Hard. We prove this decision problem is NP-Hard under randomised reductions, in four or more dimensions. In quantum information, hyperdeterminants measure quantum entanglement, under the name ``tangle''. Our reduction implies that it is hard to tell if four or more qudits are tangled, unless quantum computers can efficiently solve NP-complete problems.