  Pandharipande-Pixton-Zvonkine's proof of Pixton's generalized Faber-Zagier
relations in the tautological ring of $\overline M_{g, n}$ has started the
study of tautological relations from semisimple cohomological field theories.
In this article we compare the relations obtained in the examples of the
equivariant Gromov-Witten theory of projective spaces and of spin structures.
We prove an equivalence between the $\mathbb P^1$- and 3-spin relations, and
more generally between restricted $\mathbb P^m$-relations and similarly
restricted (m + 2)-spin relations. We also show that the general $\mathbb
P^m$-relations imply the (m + 2)-spin relations.
