You won't be improving quality, you have what you recorded. You can save the recording in 720 or 1080 depending on the hardware you choose.
I use an Elgato HDMI capture device with their software. Make sure you get one compatible with your Mac and OS.
These are real time capture devices of unprotected video. You basically connect the HDMI cable to the Mac, start up the capture software, start playing the video. Come back at the end of the show. Rinse and repeat for the next show.
You can also just record for hours and come back later and use the video editor to separate the shows.