Here is the full list of all 30 Valthirian Arc: Hero School Story trophies - there are 18 bronze, 5 silver, 6 gold and 1 platinum.
Unlock all trophies.
Obtain Valthir.
Graduate 50 students.
Unite all Five Queendoms.
Obtain Shermanstaff.
Obtain Crescent Edge.
Learn an ultimate skill.
Ascend Isabel to Mother Seat.
All facilities built at Accreditation level A.
Reach Accreditation level A.
Support one Queendom to rule the Realm.
Craft 10 weapons.
Finish Chapter 4.
Have 1,000,000 G in your Academy's funds.
Promote a student to Paladin.
Promote a student to Harlequin.
Promote a student to Arquebusier.
Craft 10 accessories.
Promote a student to Scholarsage.
Enroll 30 new students.
Finish Chapter 3.
Promote a student to Medica.
Promote a student to Arc Draconus.
Rename a student.
Finish Chapter 2.
Finish 30 missions.
Finish Chapter 1.
Buy an item from the wandering merchant.
Graduate a student.
No fallen students for 5 consecutive missions.