Chronoliths
Chronoliths are large stone structures found throughout Valisthea. You can use them to initiate the “Chronolith Trials,” which are a timed trial side quest that you can complete as outlined in the Tutorial Message that appears when you attempt to activate the first one:

You won’t be able to access the Chronoliths or activate the trials until you reach the Things Fall Apart Main Scenario Quest, but even then, it’s recommended that you save them until towards the end of the game.
Each of the Chronolith Trials is a time trial where Clive will only have access to one specific elemental Eikon. For example, in the “Trial by Fire,” Clive will only be able to access Phoenix related abilities. This limitation makes the fights much more challenging, but you can make them quite a bit easier if you tackle them after you have obtained all of Clive’s best gear, weapons, and accessories. Check out the “Best Accessories” page for some advice on how to equip Clive for these fights:
Best Accessories

This section provides recommendations on which accessories are generally considered to be the “best.” This guide can be helpful if you’re not sure which ones to use or if you want help optimizing your character setup.
You can also consider using accessories that specifically benefit one of the abilities that you have access to, and swap the accessories as Clive’s abilities change, but this can be quite cumbersome. It’s definitely possible to defeat all of the trials using the accessories on the page listed above.
The rewards you receive for completing each trial are not overly critical. In fact, the rewards for each trial are an accessory for the Eikon you were using to complete the trial. There is an achievement/trophy that you can earn from completing one of the Chronolith trials:
Trial Run
Achievement/Trophy
“Complete a chronolith trial.” Completing any one of the seven Chronolith Trials listed in the side quests section will reward you with this achievement/trophy.

There is also a curiosity that Clive receives for the Wall of Memories once he’s finished the first six trials:
For the Hoard
Achievement/Trophy
“Obtain all curiosities.” You’ll receive this achievement/trophy automatically once you’ve obtained all of the curiosities required to complete the Wall of Memories in Clive’s Chambers.

The sections below provide a map to find each of the trials and some high level tips for completing each of them:
Chronolith Trials
Trial by Fire

Activating the Hand of Hyperion (Trial by Fire) in the Greensheaves area of Rosaria.
Trial by Wind

Activating the Hand of Dione (Trial by Wind) in the Drakos Deep area of Dhalmekia.
Trial by Thunder

Activating the Hand of Iapetus (Trial by Thunder) in the Norvent Valley area of Sanbreque.
Trial by Earth

Activating the Hand of Titan (Trial by Earth) in the Northern Velkroy Desert area of Dhalmekia.
Trial by Light

Activating the Hand of Mimas (Trial by Light) in the Royal Meadows area of Sanbreque.
Trial by Darkness

Activating the Hand of Enceladus (Trial by Darkness) in the Vidargraes area of Waloed.
The Rising Tide
Trial by Water

Activating the Hand of Tethys (Trial by Water) in the Vidargraes area of Waloed.