The Time Window is a Horomancy device that allows its user to communicate with others in the past.
After Marina Andrieski and Penny escaped from the Stoppard from Timeline 40, they accidentally ended up in Timeline 36 using the Stoppard Cube. Both decided to not use the Cube until they knew how, leading Andrieski to suggest they ask the Stoppard from Timeline 36 since he had no issue with them yet.
When Stoppard's mother Sonia fainted due to her watches malfunctioning, Marina stole Sonia's notebook before she and Penny teleported away. At Timeline 36's Brakebills University, Marina read Sonia's notebook which was full of designs for devices, including the Time Window. After learning that Stoppard was pursuing them due to his mother's health failing, Penny decided to use the Window to speak to Kikuno in the past to convince her to stop using cinnabar and save herself, but Kikuno explained that her son already tried to sway her, but after learning she was about to discover a new Discipline, she decided that her work was too important.