$ACC^0$ implementation of a boolean function
Suppose $x \in \{0,\dots,2n\}$. Then we have $((2n)^2 - n) + x$ is square if and only if $x=n$. This is easily seen to imply that the square function is complete for $\mathrm{TC}^0$ under $\mathrm{AC}^0$ Turing reductions.