Where to find the hidden chests
The Leviathan is more than just a series of linked encounters to overcome. It's also a massive destination to explore in its own right—a planet-eating Cabal ship decked out in gold and purple that looks like a Prince tribute concert in space. In fact, the raid itself takes place in the comparatively tiny palace on the bow of the Leviathan, but you can also access the underbelly of the ship—not just to explore it, but to grab some hidden loot chests. These chests reward legendary engrams, Emperor Calus tokens, and frequently exotic engrams, so they're well worth putting the effort in to find.
To get into the underbelly of the ship, head to the end of the bridge you fly in on and drop off to the side. Below you'll find another, thinner bridge which leads to a small room containing three columns. There are two levers on either side of each column. These must be pulled in a specific order to open a gate up above. From left to right, number the levers one through six and pull them in the following order: one, five, three, two, four, six. This can be done solo but is much easier with a full team.
If you do this correctly, you'll see a text notification in the bottom left and the massive stone gate in the middle of the outer palace will open up. Head back outside and up, then look for a dark pit with pipes running through it. Those are your ticket inside.
There are nine hidden chests in total, but you don't have to find them all. Each time you complete one of the first three raid encounters, you'll receive a random key. Each key corresponds to a hidden chest, so after a full raid you'll only need to chase down three chests.
There are essentially two types of chests. Most are just lying around, but a few require more work. Some chests are guarded by 'watchers,' little turret-like contraptions. There are six watchers to a chest, and to open the gilded black doors that lead to these chests, you need to kill all six watchers at the same time. You don't have to be exact, but you do have to move fast. Take more than a couple of seconds and the surviving watchers will sound the alarm, locking the door and sending enemies your way. Simply locate all the watchers, sneak around them until everyone is in position, then quickly kill them all with your best power weapon. If you mess up, just exit through the nearest door, close it, and try again. Some rooms will have multiple entry points, in which case it helps to split your team up. Take a breach and clear approach and you won't have too much trouble.
Given that the chests are squirreled away in all kinds of hard-to-explain spots, we recommend using the map above (thanks, Addesso) and this video guide from YouTuber Arekkz Gaming to pinpoint the ones you need. To jump to specific chest locations, follow the video timestamps below.
Engine - 2:30
Ventilator - 4:10
Pipeline - 4:40
Conduit - 5:20
Irrigation - 7:40
Drain - 8:19
Transfer - 10:22
Armory - 11:19
Aqueduct - 12:07
Aside from bonus loot, there's one more benefit to exploring the ship's underbelly. If you're so inclined, you can use it to avoid repeating The Castellum encounter between fights. You can't access fights prematurely (so no skipping to the final boss), but you can use specific paths to access the next encounter in the rotation (which changes from week to week). This is especially helpful when you have to pull out of a raid, either due to a disconnection or to pick up where you left off on a previous day, since repeating the same Castellum round won't get you more loot.