The difference is that QA is process oriented and QC is product oriented. Testing, therefore is product oriented and thus is in the QC domain. Testing for quality isn't assuring quality, it's controlling it.
Two terms that have many interpretations because of the multiple definitions for the words assurance and control…. One definition of quality assurance is: all the planned and systematic activities implemented within the quality system that can be demonstrated to provide confidence that a product or service will fulfill requirements for quality. One definition for quality control is: the operational techniques and activities used to fulfill requirements for quality. Often, however, “quality assurance” and “quality control” are used interchangeably, referring to the actions performed to ensure the quality of a product, service or process.