The Puppet Master

After speaking with Carbuncle at the House of the Hero in Windurst Walls, head to Port Bastok and speak with Juroro @ I-8. Juroro will give you an Earth Pendulum (not a key item), which you must take to the Cloister of Tremors deep within the Quicksand Caves. Upon trading the Earth Pendulum to the Earth Protocrystal you may enter the battle field with up to 18 people. Inside you will face Galgalim, a NM Hecteyes. Upon defeating the NM you will not receive a key item nor do you need to touch the protocrystal a second time. Speak with Juroro in Port Bastok again, and then return to Windurst Walls and speak with Koru-Moru @ E-7. Koru-Moru will give you your Kukulcan's Staff!

*You must be a level 40 Summoner to activate this quest, but after activating it you may complete the quest as any other job (I am in the pic below as WHM 56).

Reward: Kukulcan's Staff
Prerequisites: Summoner of at least level 40.

Summoner Artifact Armor Quests: (this is #1 of #4)
1) The Puppet Master
2) Class Reunion
3) Carbuncle Debacle
4) Borghertz's Calling Hands