This question comes from a colleague working in econometrics. $A$ and $B$ are $n\times n$ real symmetric matrices. If we know the eigenvalues of $A$, $B$ and $A+B$, what meaningful information can we obtain about the eigenvalues of $A^2+B^2$?

I have read this related question, but here we have the extra information of the eigenvalues of $A+B$.