A complaint Friday alleged that Harvard University discriminates against Asian-American applicants by setting a higher bar for admissions than that faced by other groups.

The complaint, filed by a coalition of 64 organizations, says the university has set quotas to keep the numbers of Asian-American students significantly lower than the quality of their applications merits. It cites third-party academic research on the SAT exam showing that Asian-Americans have to score on average about 140 points higher than white students,...