Peleg--Shpilka--Volk: $\chi(|H^{\otimes n}\rangle)=\Omega(n)$

The stabilizer rank of the magic-state tensor power grows at least linearly: there are a constant $c>0$ and a threshold $N$ such that

This is the best unconditional lower bound known for an explicit state, improving an earlier bound of order $\sqrt{n}$. It is the current frontier of the mission's goal: the goal asks for a super-polynomial bound, and this result establishes the linear case. Improving it even to a super-linear bound is known to resolve a separate open question, namely the construction of a Boolean function computable in polynomial time that requires a super-linear number of summands in a decomposition into exponentials of quadratic forms over $\mathbb{F}_2$.

Formalization Note The asymptotic notation $\Omega(n)$ of the source is rendered explicitly as the existence of a positive constant $c$ and a threshold $N$ beyond which the inequality holds, with both quantified outermost. The source states the same bound for the alternative magic state $|R\rangle$; only the $|H\rangle$ case is formalized here.