  In this note we present an algorithm that lists all $4$-cycles in a graph in
time $\tilde{O}(\min(n^2,m^{4/3})+t)$ where $t$ is their number. Notably, this
separates $4$-cycle listing from triangle-listing, since the latter has a
$(\min(n^3,m^{3/2})+t)^{1-o(1)}$ lower bound under the $3$-SUM Conjecture.
  Our upper bound is conditionally tight because (1) $O(n^2,m^{4/3})$ is the
best known bound for detecting if the graph has any $4$-cycle, and (2) it
matches a recent $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ $3$-SUM lower bound for
enumeration algorithms.
  The latter lower bound was proved very recently by Abboud, Bringmann, and
Fischer [arXiv, 2022] and independently by Jin and Xu [arXiv, 2022].
  In an independent work, Jin and Xu [arXiv, 2022] also present an algorithm
with the same time bound.
