Let $X$ be a scheme and suppose that the étale cohomology $H^i(X,\mathcal{F})$ vanishes for all quasi coherent sheaves $\mathcal{F}$ and $i > 0$. Clearly if $X = \text{Spec}(k)$ for an algebraically closed field this holds. But can we also characterize the schemes for which this occurs?
Certainly there should be no nontrivial étale covers.