There are 15 Trophies that can be collected.

Bronze

Well, look what you found there
Picked up your first Collectible

In less than 12 parsecs
Achieved Goal Time on all maps

Honour Student
Achieved Shifts Goal on all maps

Did a Thing!
Used the 'Do A Thing' menu item

Nuclear Nooby
Picked up the IDS device

Dimensional Drifter
Powered up the first generator

Symmetrical Sojourner
Powered up the second generator

Peerless Perambulator
Powered up the third generator

Big Meanie
Knocked Ike over with an object

Still Alive
Made it through every level at least once without dying

Explorer of All Roads
Solved 'Choose Wisely' by choosing each battery first

Art Major
View your portrait in all dimensions

Testing the Fences
Break 20 panes of glass

Silver

Ah, finally some peace and quiet
Collected all Collectibles

Gold

Extraordinaire Extrapolator
Powered up the Uber-IDS

http://www.ps3trophies.org/game/quantum-conundrum/trophies/