It is well-known that the space or strata $\mathcal{Q}(\emptyset)$ of quadratic differentials with no poles or zeros on a genus 1 surface is empty (this is one of the four exceptions to a theorem of Smilie and Masur in Teichmuller theory which says those strata are nonempty).
My silly question is, for $dz$ the standard/classical holomorphic differential on a complex torus $T=\mathbb{C}/\mathbb{Z}+i\mathbb{Z}$, why is $w=(dz)^2$ not an example of a holomorphic quadratic differential on $T$?