There are 48 Trophies that can be collected.

BRONZE

Smooth Operator
Press 10 TARDIS console buttons

Thorough
Scan 10 objects

Head scratcher
Collect 5 Hats

Paper trail
Collect 5 Diary pages

Brimming with enthusiasm
Collect 10 Hats

Short story
Collect 10 Diary pages

Puzzler
Complete the game on puzzle medium difficulty

SILVER

Hat's off to you
Collect 20 Hats

Long story
Collect 20 Diary pages

Flying Doctor
Complete the game in 300 minutes

High IQ
Complete the game on puzzle hard

GOLD

Diarist
Collect all the Diary Pages

Mad Hatter
Collect all the Hats

PLATINUM

Time Lord
Collect all Trophies

33 SECRET

BRONZE

Crate work
Use the gold bullion crate to smash your way out of the Bank of England

Fast on your feet
Beat the Cybermen to the Underground refugee camp

On the right tracks
Defeat the Cybermen in the Underground refugee camp

Stunning work
Stun 5 Cybermen

A Plumb Job
Re-route the gas pipe plans in Victorian London

Hack Job
Open the security door in the office block

Tested to Destruction
Re-route the thermionic consoles in the Cyberfactory

The Open Road
Lower the drawbridge on Old London Bridge

No time to hang around
Collect the Stasis Field Generator in the Silence Base

Soundly Beaten
Sonic disarm 5 Silurians

Hello Sweetie
Wake up River in the tomb

Silently Silencing the Silence
Outwit the Silence without being checkpointed

Toxic Relationship
Defeat Vekkis

Goodbye Blue Boy
Outwit the Strategist Dalek

The Eyes have it
Quick-shoot 5 Daleks

Ye Doctor
Discover the statue in the Elizabethan quadrangle

Winch Way Next
Activate the crane winch in Old St. Pauls

A steal
Defeat the Silence and steal their time capsule

Hello Sexy
Find the TARDIS in the Dalek time lock

Sent packing
Send the Dalek Supreme to Antarctica

One in the eye for the good guy
Shoot the Emperor Dalek in the eyestalk

Geronimo!
Defeat the Emperor of the Daleks

Clever clogs
Complete Data Link puzzle in 60 seconds

Smart Cookie
Complete Re-router puzzle in 60 seconds

Clear Vision
Complete Mind Map puzzle on Medium or Hard in 60 seconds

SILVER

The Stormcage Redemption
The Stormcage Redemption

Pucker Up
Kiss 6 Stormcage Guards

Quick Getaway
Escape Stormcage in 8 minutes

Deleted
Finish the Cyberfactory in 20 minutes

Elementary
Finish return to Victorian London in 15 minutes

Beg for Mercy
Power-shoot 5 Daleks

Exterminated
Finish time-locked London in 20 minutes

Timey Wimey
Complete a Time Corridor in 2 minutes

Universal Destruction Avoided
Complete the game

http://www.ps3trophies.org/game/doctor-who-the-eternity-clock-vita/trophies/