Can QAP launch an application with administrator privileges?
Yes. If QAP is running with administrator permission, all applications it will launch will automatically inherit the admin level. But you should not run QAP with admin privileges unless you have good reasons to do so.
Check the option Elevate application with administration privileges in the Advanced Settings of the Edit Favorite dialog box. Of course, before launching the application, Windows will display the usual approval dialog box or ask for the admin password.
See more information about Windows ACL (Access Control List) and QAP.