jmc

The Inscribed Angle Theorem states that $m\angle PQR$ is half the measure of arc $PR$. So the measure of angle $\angle PQR$ depends only on the size of arc $PR$. The seven given points are equally spaced around the circle, so they divide the circumference into seven congruent arcs. Arc $PR$ could consist of one, two, three, four, or five of these pieces. (Draw a few quick pictures if this is not immediately apparent; in particular, convince yourself that enclosing six pieces is not an option.) Therefore there are only $\boxed{5}$ different values for $m\angle PQR$.