Does QRIS predict learning?

In a new Heartland Institute article ("Most States Use Useless Preschool Ratings, Study Finds"), Ashley Bateman writes:

Twenty-six states rate preschools with a system that doesn’t identify quality institutions, finds a new study Science journal published in August.

The Quality Rating and Improvement System (QRIS) is the most widespread preschool rating system, and it has been spreading as states increase government preschool subsidies. The problem? Researchers find QRIS doesn’t actually tell whether a particular preschool benefits kids.