Property of $Q$ such that $\frac{x^TQx}{\|x\|^2} = \text{const}, \ \ \ \forall x\in \mathbb{R}^n$

Question

I am curious about the following problem:

1. suppose $Q^TQ = I$, i.e., $Q$ is orthogonal
2. we want $$\frac{x^TQx}{\|x\|^2} = \text{const}, \ \ \ \forall x\in\mathbb{R}^n$$

My question is what properties of $Q$ to let this equality hold?

I think the only property is $Q=I$. Are there any other properties?