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Suppose we have a quantified linear program $$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$ $$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$ $$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$ $$...