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While there currently is not a lot of information regarding quests, a few things were revealed at SOE Live. Quest areas can collapse and rebuild, so that the same area may suddenly have quests players had not previously noticed. Some quests can also disappear for periods of time, likely due to the world being emergent and the destructability mechanic.
There are also Rallying Calls, which are EverQuest Next's version of public quests. The only similarity players will see with these and public quests in other games, however, is the group collaboration. Rallying Calls will be different on each server, can take 2 - 3 months to complete, will have multiple stages and those stages will require players to test out all sorts of different ideas in order to complete one stage and trigger the next. When one Rallying Call has ended, another one starts -- this could happen in a completely different area of the world or right in the same area players just completed one!