Here is the full list of all 12 Zeroptian Invasion trophies - there are 11 gold and 1 platinum.
Get all the trophies
Complete 7 levels
Destroy the boss of the second area
Complete 5 levels
Complete 3 levels
Get 10000 score
Destroy the boss of the first area
Get 5000 score
Get a power up
Get 2000 score
Get 1000 score
Die for the first time