Let's consider a random quantum circuit $C$, applied to the $n$ qubit initial state $|0^{n}\rangle$, producing the state $|\psi\rangle$.
Consider a general efficiently implementable $m$-outcome POVM measurement $\{M_i : i = 0, 1, \ldots, m-1\}$. Let
\begin{equation} p_i = \text{Tr}\big(M_i |\psi\rangle \langle \psi|\big). \end{equation}
Is anything known, in general, about
\begin{equation} \mathbb{E}[p_i] ~~\text{and}~~\mathbb{E}[p_i^{2}] \end{equation} where the expectation is taken over the choices of the random circuit?
I am especially interested in the case when the POVM elements $M_i$ describe an entangled multi-qubit measurement (which is efficiently implementable).
Note that for the special case of when the POVM corresponds to an $2^{n}$-outcome standard basis measurement, we know that \begin{equation} \mathbb{E}[p_i] = \frac{1}{2^{n}}, ~~~ \mathbb{E}[p_i^{2}] = \frac{2}{2^{n}(2^{n} + 1)}. \end{equation}