Let's assume $X$ is connected with friendly basepoint. The quick answer goes as follows. Imagine all subgroups $\Sigma_I < \Sigma_n$ of symmetric groups formed from smaller symmetric groups by iterating products and wreath products. For such an $I$, let $$D_I(X) = E\Sigma_{n+} \wedge_{\Sigma_I} X^{\wedge n}.$$
Then $Q^{\infty}(X)$ will be the product over all $I$ of the spaces $QD_I(X)$. As other folks have already hinted at, this follows from the stable splitting of $QX$.

By the way, the cosimplicial resolution $X \rightarrow Q(X) \cdots$ was used by Gunnar Carlsson to deduce the Sullivan conjecture from the Segal Conjecture, and by Greg Arone and Marja Kankaanrinta to give a second construction of the Goodwillie tower of the identity.

Dually, if $X$ is an infinite loopspace, one gets a simplicial infinite loopspace $X \leftarrow QX \cdots$. Back in the day, I thought about this quite a bit. When $X=S^1$, the Whitehead conjecture sequence formally splits off this. For general $X$, I proved a splitting result about the beginning of this in [Topology 23 (1985), 473-480]. This has a fun conceptual proof (no homology calculations!), works for all primes $p$, and includes Finkelstein's $p=2$ theorem and the Kahn-Priddy theorem at all primes. I would love to see someone generalize this in some interesting way.