Many bounded versions of undecidable problems are NP-hard

Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and membership for quantum correlations. Most of these results rely on reductions from a handful undecidable problems, such as halting problem, tiling Post correspondence or matrix mortality problem. All common property: they an NP-hard bounded version. This work establishes relation between unbounded their versions. Specifically, we show that NP-hardness version follows easily reduction problems. leads to new simpler proofs positivity product operators, reachability ground state energy sheds light intractability in theoretical physics computational consequences bounding parameter.