Quality Care for Moms and Babies Act This bill directs the Centers for Medicare & Medicaid Services (CMS) to: identify and publish a recommended core set of maternal and infant quality measures for women and children, as specified by the bill; publish an initial core set of any such measures applicable to mothers and infants eligible under Medicaid or the Children's Health Insurance Program (CHIP); establish a Maternal and Infant Quality Measurement Program; and establish an online clearinghouse of resources for entities working to improve maternity and infant care quality. The CMS may make grants to eligible entities for: the development of new state and regional maternity and infant care quality collaboratives, expanded activities of existing collaboratives, and maternity and infant care initiatives within established state and regional quality collaboratives that are not focused exclusively on maternity care. Under current law, the CMS must contract with a consensus-based entity to carry out specified duties regarding performance measurement. The bill requires such an entity to facilitate increased coordination and alignment between the public and private sector with respect to quality and efficiency measures.