We propose a projection-based class of uniformity tests on the hypersphere using statistics that integrate, along all possible directions, the weighted quadratic discrepancy between the empirical cumulative distribution function of the projected data and the projected uniform distribution. Simple expressions for several test statistics are obtained for the circle and sphere, and relatively tractable forms for higher dimensions. Despite its different origin, the proposed class is shown to be related with the well-studied Sobolev class of uniformity tests. Our new class proves itself advantageous by allowing to derive new tests for hyperspherical data that neatly extend the circular tests by Watson, Ajne, and Rothman, and by introducing the first instance of an Anderson-Darling-like test for such data. The asymptotic distributions and the local optimality against certain alternatives of the new tests are obtained. A simulation study evaluates the theoretical findings and evidences that, for certain scenarios, the new tests are competitive against previous proposals. The new tests are employed in three astronomical applications.