What type of PCR quantifies the rate at which new DNA is synthesized?

It's Real time. Just as the name implies, real-time PCR is the type of PCR that quantifies the rate at which new DNA is synthesized. Real-time PCR monitors the progress by using a florescent dyes reporter molecule. The fluorescent emitted by the reporter allows early detection of PCR production early in the reaction.

Compare to the traditional PCR; real-time PCR is less time consuming due to the ability to detect amplification early in each cycle. Real-time PCR does not require post PCR processing. Real-time PCR can be used in all applications, just like the traditional PCR mostly in biological science, genetic expression analysis, and genetic testing.