When you are watching a video in QuickTime’s full screen mode and you use cmd⌘ + tab to switch to another application, the movie will automatically exit full screen mode. To stay full screen in the background, use the following defaults write command:
defaults write com.apple.QuickTimePlayerX MGFullScreenExitOnAppSwitch 0
To make the command take effect, you need to restart QuickTime.
Revert to the default setting (manual play) change value 1 in 0.
defaults write com.apple.QuickTimePlayerX MGFullScreenExitOnAppSwitch 1