Disgaea 3: Absence of Detention

HQ is easily raised by playing the same map over and over. You’ll probably get this while power levelling. In the process of grinding out 7-3, keep playing it over and over and this will unlock once your HQ hits ~300, this will unlock.

There's also some sort of feature built into the Vita GPS that’s supposed to raise your HQ. If anyone has any info on how exactly this works, let me know. I recently took my Vita on a trip from Boston to Atlanta, with the GPS enabled, and didn’t notice any change in my HQ the entire time.