The SQ are a mysterious organization which acts as the main enemy in the books. It seems to act very different in the past and present however.
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The SQ in the present, are an all knowning group that has almost total control over the major governments of the world. Having had its hand in just about every major historical event, the SQ's have total power. Sera's uncles are feared by many but others act as though they have a lot of respect for the SQ. They even run history museums like the Smithsonian. To be a part of the SQ's are considered a high honor, there is even some kind of reality TV show competition where the winner is declaired the next SQ's intern. The SQ's are still secretly fighting the Hystorians and trying to prevent them from getting time travel.
Recently revealed in a post on the Infinity Ring forums that the past SQ's remained secretive until World War II, after which they became bolder and stepped out of the shadows (possibly taking the credit for what they have done over the years in secret). It was also revealed that Aristotle (founder of the Hystorians) was the first casualty in the war between the 2 groups. It also seems that the Lady in Red is the current leader of the SQ in modern times. It is still unknown however what exactly SQ stands for.
The SQ and their Great Breaks are the reason why the 2012 in the first book is very different from our own. It can be assumed that once the Breaks are fixed that, 2012 will look like our own and the SQ will either be non-existant or still exist but have no power.