First introduced in The Legend of Heroes: Ao no Kiseki and carried onward from the Trails of Cold Steel trilogy. They are installed in the center slot through new battle orbments such as the ENIGMA and the ARCUS.
Master Quartzes can level up from 10% of the equipped character's EXP from battle. Between levels 1 (default) to 5 (maximum), they gain increased stats and sometimes grant new Orbal Arts and abilities.
Ao no Kiseki (+EVO)
When a Master Quartz reaches level 5 MAX, its respective elemental Master Arts is unlocked. Probably only usable in the ENIGMA II orbments.
Trails of Cold Steel (I + II)
Probably only in the ARCUS modules, the Master Arts have been subsided as the equipped user gains extra arts and abilities when the MQ is leveled up.
Trails of Cold Steel III
In the newest ARCUS II, there are two Master Quartz slots, allowing the user to equip two at once and increasing their maximum level to 7. The top-main and bottom-sub slots divided the amount of acquired experience where 100% goes to the main, while 50% goes to the sub.