The Hall of Darkness hunt is a QR side quest. As the name implies, your party will be taking part in a hunting competition. The hunt takes place on B2 of the Hall of Darkness. A soldier will tell you the details of what will be occurring and what monsters you are to be killing.
When you enter the floor, a soldier will hail you. He tells you that there different levels in this hunt. The first level is focused on you hunting Muskoids and Hollow Magus. You are then told that the ranking is based off of how many of each you kill while on this floor. If you leave this floor at all before the match is complete, your tally becomes fixed for the remainder of the tournament. (It may be beneficial to have healing, revival, and status removing items before handling this quest.) If you have less than 10 as your tally, you have considered to have failed the request. When you are done, head to the guild master for your ranking.