Suppose we define a magic state on $2n + 1$ qubits as: \begin{equation} \prod_{i = 1}^{n} CCZ(0, i, i+n) |+\rangle^{\otimes 2n + 1}. \end{equation}
Does anyone have an intuition for the scaling of the amount of magic in such a quantum state in $n$ with respect to some monotone? As a comparison, the $|CCZ\rangle^{\otimes \Theta(n)}$ tensor product of magic states clearly has $\Theta(n)$ magic measured by the family of $\alpha$-stabilizer Renyi entropies (https://arxiv.org/pdf/2106.12587) by additivity.
