Here is the full list of all 15 The Last Cube trophies - there are 10 bronze and 5 silver.
Complete all Steps and Time Challenges.
Complete all bonus levels.
Complete 4 challenges.
Complete all challenges.
Find all relics.
Find all Relics in the Hub.
Stack 3 Clone Cubes on top of each other.
Find the lone robot arm in the Research Lab.
Walk onto a Full Reset Square while having one of each sticker on the Cube.
Complete the Blue Theme.
Complete the Green Theme.
Complete the Orange Theme.
Complete the Purple Theme.
Complete the Red Theme.
Complete the Yellow Theme.