If a morphism between schemes $f : X \to Y$ is both quasicompact and quasiseparated, could we prove that $f : X \to Y$ is a separated morphism?
In other words, if $f : X \to Y$ is quasicompact and $\Delta : X \to X \times_Y X$ is quasicompact, could we prove that $\Delta : X \to X \times_Y X$ is a closed immersion?
The affine line over a field with doubled origin already provides a counterexample. This scheme is noetherian, hence every subset is quasi-compact, hence any morphism out of it is qcqs.