Let $P,Q$ be two real symmetric matrices of dimension $n \times n$. Define two sets $A=\{x \in \mathbb{R}^n: x^T P x \le 0 \}$ and $B = \{ x \in \mathbb{R}^n: x^T Q x \le 0 \}$. If $A \subset B$, then what's the relationship between $P$ and $Q$? For example, if $Q$ or $Q-P$ is semi-negative definite, then the inclusion is always true. But what's the equivalent characterization, possibly by a linear matrix inequality?
My own thoughts: the two sets $A$ and $B$ define two cones. Then $A \subset B$ should be equivalent to the boundary of $A$ lying in $B$. But still, I don't know how to translate this as a matrix inequality.