Stealing the Past - Lestallum
First Available: Chapter 15
Other Requirements: Complete the game and obtain flying Regalia
Location: Lestallum
Rewards: 50,000 EXP
This quest starts in Lestallum any time after you've completed the game and unlocked the flying Regalia. What you need to do is head to the eastern most portion of Lestallum (pictured above and below) and eavesdrop on a conversation. It has to be at night (any time after 21h but before 4h).
If you're having trouble locating the window in Lestallum, check out my video Where to find the window for Stealing the Past in Lestallum. Or, just start at the weapon merchant in eastern Lestallum and follow the path east. As it wraps around the building to the south; the window will be on your right.
Stealing the Past (Quest Start) Map Location.
Once you've obtained the quest it's time to head down south to the Ravatogh region. There is a hidden airstrip here that you need to land on (pictured below) with the Regalia. It's a very tough landing and it may take you a few tries to successfully pull off. I recommend decreasing your speed as much as possible and start to descend before you reach the runway; watch how a real plane lands on youtube if you're confused.
After you've successfully landed you'll want to travel northwest to the campsite outside of Pitioss Ruins (pictured on my map below).
Note: I strongly recommend resting at the campsite near Pitioss before completing this quest and eating the best EXP boosting food you have. One of the easier EXP boosting foods to obtain is the Stacked Ham Sandwich from Wiz Chocobo Post which boosts EXP earned by 50%. With that, this quest will give you a total of 75,000 EXP. Then you can rest at Galdin Quay for 2x more EXP and get a total of 150k just from this quest!
As for completing the quest, there are two possible locations that can contain the Magitek Core that we need. When you arrive at Pitioss Ruins there is a dip that you have to jump across outside of the dungeon and right after you do, your party leaves Noctis on his own for the dungeon. In that dip you'll sometimes find the Magitek Core that we need for the quest. This is where I found mine.
However, sometimes the Magitek Core spawns just outside the entrance of Pitioss and to the right in a corner. Before you jump through the broken gate to enter Pitioss just to your right you'll see some rubble on the ground, sometimes the Magitek Core spawns here.
It's recommended that you hold onto the Magitek Core as it's a component in upgrading the Drillbreaker through Cid.