Lost Tools
You can start this quest when you reach the Viletian army camp area by talking to Neliste any time after your conversation with Haristel. The alchemist will ask you to recover their misplaced Alchemy Tools, which can be found in the Lake Serene - 38 AE area, in the checkpoint room just before the Azure Queen boss. Return to Neliste with the tools and you'll be rewarded with an Essence Crystal, and Neliste will now be able to reinforce orbs.
Give Neliste her alchemy tools (left) and she'll be able to reinforce orbs (middle) after which talk to her again to start the next quest, "Our Fate" (right).
Our Fate
You can start this quest immediately after finishing "Lost Tools", just talk to Neliste again and she'll ask you to use your ability to time travel to find out how the war between Vilete and Lachiem ended. Easy enough, just return to the Varndagray Library - 1062 AE gate and talk to the Librarian, who will give you the Security Keycard V relic. Head up some stair to the right (from the Librarian) and go through the "V" Security Door and Lunais will automatically read the terminals therein. Once done, just return to Neliste to finish the quest.
Finding a Cure
This quest may well start after you rescue Rameda from the Caves of Banishment, but we suggest waiting until after clearing the Royal Towers - 38 AE area as well. Once those are done, return to Neliste and she'll use Eschem's condition as an excuse to ask you to see if the future Lachiemi found a cure for the bleakness. To do this you'll ned the Security Keycard B relic, which you can acquire by heading to the Sealed Caves - 1062 AE area and looting a skeleton outside of a sealed "A" Security Door. The Sealed Caves start at the southwestern end of the Lake Desolation - 1062 AE area.
Once that's in your possession, travel back to the Military Hanger - 1062 AE gate, return to the nearby elevator and ride up to the "B" security door, beyond which you'll find the terminal you need. When you have the info, return in to Neliste to complete this quest and talk about what happens with the bleakness over the next ten centuries. Interesting stuff for sure, and you will be rewarded with a Gold Necklace.
Talk to Neliste and she'll ask you to acquire some Plasma Pod Cores so she can study them (left) which you can gain by farming Plasma Pods (middle). When you have three Plasma Pod Cores, return to Neliste (right).
Exotic Imports
This quest will be available after you complete Neliste's last request, so talk to her again and she'll ask you to bring her three Plasma Pod Cores. You can find the Plasma Pod enemies who drop Plasma Pod Cores in the upper heights of the Royal Towers area, particularly in the eastern tower. Grind them until you get the three Plasma Pod Cores you need. When you have what you need, give the requested items to Neliste. She will end up rewarding you by giving you some Elemental Beads and sharing some personal info about herself.
Use dash and double jump to clear the pit (left) to find a Galaxy Stone in a chest (right).
Sea of Stars
This quest requires you to have completed all other quests, first, including Haristel's "Defector Duty" and Seykis's "Meat Box" quests. Once those are done, talk to Neliste and she'll ask you to find her a Galaxy Stone. You'll find this object in the Sealed Caves - 1062 AE area, just jump the large pit in this dungeon and loot a chest to get your Galaxy Stone. Return to Neliste and you'll get a lengthy scene, after which Neliste will give you the Galaxy Earrings item.