*Missable if you’re on the Good Ending*
In order to advance quest rank, you’ll first have to unlock Advancement Tests, and complete them. You unlock those by completing enough quests. Each rank quest has a hidden point value, and obtaining enough points unlocks an Advancement Test. The higher the quest rank, the more points it gives, so do quests of the highest available rank to unlock them faster. You can repeat the same quest as many times as you want and it’ll still contribute towards unlocking the Advancement Test. The trophy unlocks once you turn in the final advancement quest. The list of advancement tests are below.
Advancement Test 1: Defeat Phoenix (found in Bandicrash, Lastation)
Advancement Test 2: Obtain Magical Wolf Fang (defeat Fenrir, found in Castle Exterior, Lowee)
Advancement Test 3: Defeat Turtle (found in Zega Forest, Leanbox)
Advancement Test 4: Defeat Plaid Dolphin (found in Ario Plateau, Lowee)
Advancement Test 5: Defeat Vestiges of Momus (Tough Foe found in Graphic Pass, Eden)
A few quick quests to repeat to reach Quest Rank S are: "Stress relief" (defeat 8 Tetrisi), and "A gas trap" (defeat 8 Cass Traps). These are Rank A quests, and the enemies can be found in the discovered dungeon Graphic Pass in Eden. The enemies are easy to defeat at this stage of the game.