An n-dimension quadratic cone is defined as $$Q^n=\left\{x\in\mathbb{R}^n:x_1\geq\sqrt{x_2^2+x_3^2+\cdots+x^2_n}\right\}.$$
I read that $||x||^2_2\leq y$ if and only if $(1/2, y, x)\in Q^{n+2}.$ This statement looks very confusing to me. $(1/2, y, x)\in Q^{n+2}$ seemingly means $$\frac12\geq \sqrt{y^2+||x||^2_2}.$$ How is this related to $||x||^2_2\leq y$?
This does not seem to be true. The exact statement is (see e.g. here):
$ \left\| x \right\|^2_2 \leq y \Leftrightarrow (1/2, y, x) \in \mathcal{Q}_r^{n+2}$, where $\mathcal{Q}_r^n$ is the rotated quadratic cone, given by
$$ \mathcal{Q}_r^n = \left\{ x \in \mathbb{R}^n \ \middle|\ 2x_1 x_2 \geq x_3^2 + \dots + x_n^2 \right\} $$
In this case, both directions are easy to verify.