  In this paper, we give some new genus-3 universal equations for Gromov-Witten
invariants of compact symplectic manifolds. These equations were obtained by
studying new relations in the tautological ring of the moduli space of
2-pointed genus-3 stable curves. A byproduct of our search for genus-3
equations is a new genus-2 universal equation for Gromov-Witten invariants.
