A Quest of Spirit is a trial undergone by the characters in Problem Sleuth. Each of the four main characters embarks on a quest for one of the four kingdoms, involving them to commit some type of travesty upon another kingdom, further exacerbating the plight of the four kingdoms.
Successfully completing a Quest of Spirit earns you a boon from the kingdom you completed it for. This allows you to pass through the corresponding bust in the pentagram room, and allows you to become aware of your Female Alter Ego, provided you have sufficent Imagination.