I am interested in computing the sum of squares of determinants of principal minors. Let $A$ be an $n\times n$ positive semidefinite matrix and $A_S$ be a principal minor of $A$ indexed by the set $S \subseteq \{1,\ldots,n\}$. The classical result (without squares) is:

$\sum_{S \subseteq \{1,\ldots,n\}} \det(A_S) = \det(A+I)$

Are there any results on computing

$\sum_{S \subseteq \{1,\ldots,n\}} \det^2(A_S)$

or any other powers?