  We prove a new sufficient pair degree condition for tight Hamiltonian cycles
in $3$-uniform hypergraphs that (asymptotically) improves the best known pair
degree condition due to R\"odl, Ruci\'nski, and Szemer\'edi. For graphs,
Chv\'atal characterised all those sequences of integers for which every
pointwise larger (or equal) degree sequence guarantees the existence of a
Hamiltonian cycle. A step towards Chv\'atal's theorem was taken by P\'osa, who
improved on Dirac's tight minimum degree condition for Hamiltonian cycles by
showing that a certain weaker condition on the degree sequence of a graph
already yields a Hamiltonian cycle. In this work, we take a similar step
towards a full characterisation of all pair degree matrices that ensure the
existence of tight Hamiltonian cycles in $3$-uniform hypergraphs by proving a
$3$-uniform analogue of P\'osa's result. In particular, our result strengthens
the asymptotic version of the result by R\"odl, Ruci\'nski, and Szemer\'edi.
