Let $T$ be a symmetric tensor in $\left( \mathbb{R}^d \right)^{\otimes n}$ such that the polynomial $$\sum_{i_1,i_2, \ldots i_n}T_{i_1i_2 \ldots i_n} x_{i_1}^2 x_{i_2}^2 \ldots x_{i_n}^2$$ is a sum of squares polynomial. Clearly, the set of all such tensors form a convex cone, which we call $\Sigma^n_d$. I want to know if there is any description of the the dual cone of these set of tensors, i.e., $$(\Sigma^n_d)^* = \left\{ Q \in \operatorname{Sym}((\mathbb{R}^d)^{\otimes n}) \mid \langle Q,T\rangle \geq 0 \right\}$$ where $\langle Q,T\rangle = \sum\limits_{i_1,i_2, \ldots i_n}Q_{i_1i_2 \ldots i_n}T_{i_1i_2 \ldots i_n}$ is the Euclidean inner product.
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