Here is the full list of all 92 Idle Gem Quest achievements.
Activate 5000 Runes as the Mage
Play 10000 Magic Notes as the Bard
Actively Attack 250 times while on fire
Play 5000 Magic Notes as the Bard
Defeat 16 Monsters
Spend 250 Diamonds
Upgraded the Ninja to level 10
GPS of over 1 Pointy Emerald gem per second
Upgrade the Mage to level 7
Play the Monster hunt 100 times
Spend 100 Diamonds
Activate 1000 Runes as the Mage
Obtain a Huge Green gem
Activate 500 Runes as the Mage
Spend 50 Diamonds
Play the Monster hunt 50 times
Play the Monster hunt 25 times
Spend 10 Diamonds
Activate 100 Runes as the Mage
Cast 10 perfect 'hard' spells in a row
Land 10 arrows in a row without touching the ground
Activate 10 musical notes in 1 attack
Perform 100 slices in a row
Play the Monster hunt 5 times
Play 1000 Magic Notes as the Bard
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Upgraded the Archer to Level 12
Upgraded the Bard to Level 15
Play 500 Magic Notes as the Bard
Slice monsters 10000 times as the Ninja
Upgraded the Warrior to Level 15
Upgrade the Mage to level 5
Upgraded the Ninja to level 7
Defeat 12 Monsters
Upgrade the Mage to level 3
Attack 100 times in 10 seconds
GPS of over 1 Blood Shard gem per second
Unlock the Mage
Obtain a Scum Green gem
Upgraded the Archer to Level 8
Upgraded the Bard to Level 10
Slice monsters 5000 times as the Ninja
Upgraded the Ninja to level 4
Upgraded the Warrior to Level 10
Unlock the Ninja
Defeat 8 Monsters
Slice monsters 1000 times as the Ninja
Slice monsters 500 times as the Ninja
Play the Monster hunt 1 time
Slice monsters 100 times as the Ninja
Upgraded the Archer to Level 4
Play for over an hour
Send 10000 Arrows at monsters as the Archer
Actively Attack 1000 times while on fire
Send 5000 Arrows at monsters as the Archer
Send 1000 Arrows at monsters as the Archer
Send 500 Arrows at monsters as the Archer
Send 100 Arrows at monsters as the Archer
Obtain a Blood Red gem
GPS of over 1 Spiky Green gem per second
Attack 10000 times with the Warrior
Unlock the Archer
Play 100 Magic Notes as the Bard
Upgraded the Bard to Level 5
Defeat 4 Monsters
Upgraded the Warrior to Level 5
Attack 5000 times with the Warrior
Obtain a Perfect Emerald gem
GPS of over 1 Smoky Purple gem per second
Unlock the Bard
Attack 100 times with the Warrior
Attack 500 times with the Warrior
Attack 1000 times with the Warrior
Unlocked the Warrior
Max out the Warrior
Max out the Bard
Max out the Archer
Max out the Ninja
Max out the Mage
Activate 10000 Runes as the Mage
Actively Attack 100 times while on fire
Obtain a Priceless Max gem
GPS of over 1 Lime Dice gem per second
Defeat ALL Monsters
Spend 500 Diamonds
Own 1 Artifact Gem
Own 2 Artifact Gem
Own 3 Artifact Gem
Own 4 Artifact Gem
Own 5 Artifact Gem