Here is the full list of all 36 Valthirian Arc: Hero School Story 2 trophies - there are 17 bronze, 14 silver, 4 gold and 1 platinum.
Obtain all trophies
Be a Principal for 10 months
Complete 5 Quests
Complete 5 Research Projects
Defeat 50 Monsters
Graduate 5 Students
Play the Eating Minigame 4 times
Play the Fishing Minigame 4 times
Play the Snowman Minigame 4 times
Play the Swimming Minigame 4 times
Send 7 Students on Errands
Spend 10000 Gold
Spend 500 Arcstones
Use 10 Skills in combat
Win the Eating Minigame 5 times
Win the Fishing Minigame 5 times
Win the Snowman Minigame 5 times
Win the Swimming Minigame 5 times
Achieve Maximum Relations with All Factions
Be a Principal for 500 months
Complete 100 Quests
Complete 120 Research Projects
Defeat 5000 Monsters
Graduate 60 Students
Have 150,000 Gold before 1305
Play the Eating Minigame 20 times
Play the Fishing Minigame 20 times
Play the Snowman Minigame 20 times
Play the Swimming Minigame 20 times
Reach 4000 Academy Prestige Points
Send 750 Students on Errands
Spend 20000 Arcstones
Spend 5000000 Gold
Unlock the Sky Castle
Use 5000 Skills in combat
Defeat Fellwing and finish the main story