Adil Hussain mentions the Temporal War that happened before the fall. In the first three seasons of Star Trek: Enterprise (2001), agents from a "temporal cold war" repeatedly interfered with Jonathan Archer and the crew of the NX-01 Enterprise.
This episode, set in 3188, is the farthest in the future of any Star Trek episode where the approximate year is known, surpassing Living Witness (1998) which was set around 3074.
Grudge appears to be a normal Terran Maine Coon cat.
The episode was originally announced under the title "That Hope Is You: Part 1". The addition "part 1" was removed shortly before the second episode "Far from Home" was broadcast, as this episode had an independent title without the addition of "part 2". When the title for the third season finale was announced ("That Hope Is You: Part 2") the addition "part 1" was again added to the episode.
One of the couriers hunting down Book and Burnham is a Lurian, the same species as fan-favorite character Morn on Star Trek: Deep Space Nine (1993).