Moving between 3-manifold triangulations is NP-hard

We show that $\text{textsc}numberofbistellarmovesandsparsedegree-twoedgecollapsesfor3-sphere$ is NP-hard. It follows that a similar problem for an arbitrary 3-manifold is NP-hard as well. This is the first NP-hardness result concerning moves between two triangulations of a 3-manifold.