Cassiopeia, a pocket watch given to Negi by Chao Linshen allows him to time travel. Only three have been shown to exist so far, with Chao at two and Negi with one. The amount of magic used is the determining factor of how far one can travel, according to Chao in the manga, 24 hours is the limit for a single mage, and is only demonstrated to allow travel in time, not space (although in the anime it was used to take the entire class to Germany in the year 1993). Negi used the device during the Mahora Festival, in order to keep his promises to visit each of his students' festival activities.
Unfortunately, the watch is also vulnerable to sabotage - when it was brought into Evangeline's resort, a trap that Chao Linshen had built into it caused the resort to keep them for a week in the real world when it should have only been 24 hours. Afterwards, it became inactive since the main source of energy was the World Tree during the Mahora Festival. Negi and the gang were able to reactivate the watch by going deep into the core of the World Tree, which still had some residual magic left for a time jump. Apparently, it was enough to make it back to the third day of the Mahora Festival, but the watch was damaged at the end of the jump when Negi collapsed and dropped it, implying that its further usefulness would be limited.
During the "Mahora Mage Order vs. Mars" mock battle, Negi managed to use the Cassiopeia to counter the time-displacement effects of Tatsumiya Mana's bullets. Later, during the climatic battle against Chao Linshen, Negi used a pair of specially-empowered spirits in order to make minute adjustments to his own Cassiopeia and keep pace against Chao's time-jumping attacks through her Battlesuit Cassiopeia. Chao had no idea that Negi, through his genius and absolute mastery of the spells, was able to do this and was chagrined when he informed her. In the end, the damaged Cassiopeia self-destructed from over-use, but not before Negi destroyed Chao's.
Even if the watch designs are the same, Cassiopeia No.2 is carried by Chao and allows her to go into the future if the time-watch is already filled with magic power. She used it to escape from the teachers after the Mahora Fighting Tournament half-a-day. Like Cassiopeia, it does not allow her to travel in space. No details have been said about the limits. Chao uses this one to return to her own time at the end of the Festival.
Cassiopeia No. 3
For the battles in the second and final day of the festival, Chao's Battlesuit has a third Cassiopeia unit built into the back of the suit, which is controlled by an AI computer. The Cassiopeia unit enables her to circumvent attacks by moving to a different time period or simulate time stopping by continuously jumping to the same time period in a different dimension, thereby giving her the illusion of teleportation. She uses these techniques to take out Negi and escape from holds by both Setsuna and Kaede on the second day. On the third day, she takes out Asuna and Setsuna using the suit and sends Takahata into the future with her time-displacement bullets. However, during her fight with Negi he manages to destroy the unit with a well-placed punch, thereby preventing her from using it to skip through time. It is not clear exactly how the unit is operated since unlike the other units the suit has no clear setting or triggering mechanism. It would likely be either manipulated through movements of her gloves or through a neural interface. Both are entirely possible due to both Chao's and Satomi's genius.