Just play and complete past level 5 in action mode for the trophy.
TIPS: If you make a Hyper Cube, DO NOT USE IT. Make other moves around and TRY NOT BLOWING THE HYPERCUBE UP so if you have no more moves you can use the Hypercube on the gem that has the most on the screen. Count them and whatever you have the most of take them away. If you run out of moves you have too start over.

Just play and complete past level 5 in action mode for the trophy.
*WARNING: As you make it in higher stages close to 5, the bar you fill up on the bottom from matches will drain slowly until you progress. It goes a bit quicker...you gotta be quick with your fingers and spot matches quickly to beat it from getting to the end.
HINT: If you got a hyper cube early on and you're pretty quick with your fingers, try saving the cube for Level 5 unless you really need too use it. If you can blow up the power gems as soon as you can it can cause more gems will fall and may just make matches and allow the bar to fill up quicker.

No worries here just play until you complete level 10. You cannot lose; this mode is endless and you cannot run out of moves so it's one of the easiest trophies.

I happened to get this while playing normally. This is pretty easy if you get power gems and you make an explosion and another 1 or 2 power gems are near by that can be hit to cause a chain reaction.

*Tip by SwatTJ* This trophy is simple. All you must do is match only blue gems for 3 turns consecutively. This simply means match 3+ blue gems, then match 3+ more, and find 3+ more once again. Perhaps try to match only non-blue gems in Endless Mode so that you can set up many matches for blue gems.

If you sit and play endless mode for some time you will get this trophy eventually so it's pretty easy and straight forward.
If you play classic mode and run out of moves, the next screen to pop up that will tell you how many Hypercubes you made, your biggest cascade, how many power gems have been made, and how many gems cleared.

I made it too Level 16 in Classic Mode and i got 130 something thousand in total. Like I said earlier, if you have hyper cubes don't use them unless you have to. I think doing this in action mode is harder but if i remember right on the 6th stage in action mode I think I got over 100,000 points.

Just play it normally. If you're worried about running out of moves, play in Endless Mode. If you make a match and gems fall to take other gems away, those count as cascades so if you make 6 in a row you got the trophy.

Just make a match of 4 gems to create a Power Gem. Do this 200 times and the trophy is yours for the taking. Power gems can be tracked in the statistics screen mentioned in the "Jewel Keeper" trophy.

Just make a match of 5 gems to make a hyper cube and do this 50 times while playing to get the trophy.
TIP: This and the Power Gem trophy are easiest to get in Endless Mode. You can play without running out of moves so it's just a matter of time.

One of the challenging trophies in the game. It is probably best to be done in Endless Mode where you have to have 5 Hypercubes at once on the screen and the only way to do that is if you make one, not use it and save it and try to make another along the way. If you run out of moves then you may have to use one.
TIP: You can enter the XMB and quit out of the game once you have 2, 3, or 4 hypercubes made. You can decide how often to do this as maybe having 2 on screen doesn't make you worried and you don't feel the need to save.

Complete all 80 levels in Puzzle Mode, some are challenging while most are easy.
TIP: If you're stuck on a puzzle, look up a guide found on various sites on the internet. Also there are unlimited hints that can be obtained by using the in game hint system. There is one level that you cannot use hints on and here is a video showing how to complete it.