Quality Higher Education Act of 2021 This bill revises the accreditation standards under which an accrediting agency or association must assess an institution of higher education (IHE), and it requires the Department of Education to establish procedures for an agency or association to follow when an IHE fails to meet these standards. An accrediting agency or association must measure an IHE's success in student achievement against at least one standard each in the categories of completion, workforce participation, and progress toward meeting such completion and workforce participation standards. An agency or association must also make certain information such as student achievement standards and IHE performance benchmarks available on a publicly accessible website. Additionally, the National Advisory Committee on Institutional Quality and Integrity must evaluate the effectiveness of the revised standards and compare similar accrediting agencies or associations based on their assessments of IHEs.