Mrazović and Kovač, and independently Alon, have observed that the existence of some valid choice of $Q$ follows easily from Vinogradov's theorem that every large odd integer is the sum of three distinct primes. In particular, there exists some $N$ such that every prime $>N$ is the sum of three distinct (smaller) primes. We may then take $Q_0$ to be the set of all primes $\leq N$ (in which case all primes are eventually in some $Q_i$).