Temporal science is the field of advanced science involving the study of the workings of time and its effects on the space-time continuum. The understanding of temporal mechanics and temporal physics were particularly relevant for understanding time travel. Temporal mechanics was among the course material taught at Starfleet Academy.
Miles O'Brien was not particularly fond of the paradoxes created by time travel, as was the case in 2371, when he encountered a past version of himself, and they became confused as to why the past O'Brien was affected by radiation but his future self felt fine. After mulling over it for a moment, they both declared that they "hate temporal mechanics". (DS9: "Visionary")
Julian Bashir and O'Brien both took an Elementary Temporal Mechanics course when they attended the Academy. After having time traveled back to 2268 and having a chance encounter with a Lieutenant Watley aboard the USS Enterprise, Bashir was convinced that he may be experiencing a predestination paradox, and that he "could be destined to fall in love with that woman and become my own great-grandfather." (DS9: "Trials and Tribble-ations")
Both Kathryn Janeway and B'Elanna Torres knew the "finer points" of temporal mechanics, which included understanding one of the more difficult concepts of the field to grasp, which was that "sometimes effect can precede cause" and that "a reaction can be observed before the action which initiated it." (VOY: "Parallax") However, Seven of Nine was the versed in temporal mechanics aboard the USS Voyager. (VOY: "Shattered") While under the guidance of Janeway in 2378, Q, Junior was scheduled for lessons in the study of temporal mechanics. (VOY: "Q2")
The Krenim had an advanced understanding of temporal mechanics, as they possessed deadly temporal science based weapons. One such Krenim, named Annorax, was responsible for creating a temporal weapon ship. (VOY: "Year of Hell") After being taken prisoner by the Krenim, Chakotay was described by Annorax as seeming to understand the subtleties of time. (VOY: "Year of Hell, Part II") After Seven of Nine experienced a number of personalities in 2375, as a result of being linked to Borg vinculum, she manifested the personality of a Krenim scientist with whom debated "the finer points of temporal physics" with Captain Janeway. (VOY: "Infinite Regress")
Kurros, the spokesperson of the "Think Tank", described the jellyfish-like lifeform, another member of the Think Tank, as the "resident expert on temporal physics" and "perhaps the most gifted member of our group." (VOY: "Think Tank")