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 $.