A battle trophy is a certain condition that you must perform during battle to collect the trophy. Completing the condition will add the date and time of the completion to the list of all completed trophies. To view them simply go to the main menu and choose battle trophies. In order to collect these trophies you must create the data at the battle simulator at the beginning of the game. Starting a new game and loading the same battle trophies data will allow you keep all of the stats from your last game. Here are the following things you can unlock for completing a certain percentage of battle trophies.

15%(45 Trophies):Allows the use of each player's 2P costume.
25%(75 Trophies):Allows the use of the Universe difficulty mode.
40%(120 Trophies):Allows the use of each player's 3P costume.
55%(165 Trophies):Allows the use of music test.
65%(195 Trophies):Allows the use of 4D difficulty mode.
80%(240 Trophies):Allows the use of each player's 4P costume.
95%(285 Trophies):Allows the use of Full Active Mode.

Refer to the Battle Trophies List for more details.