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).

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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}