What Was Changed From The Comics For The Show?
While the name of the Ninth Circle has yet to be dropped on Arrow, their influence is plain to see. The base concept of a sinister financial group with enough power to control governments and corporations, as well as one led by a man named Dante, is too similar not to be a nod to the villainous organization from the Green Arrow comics. It's also worth noting that the Ninth Circle employed Emiko Queen's mother (the assassin Shado) in the comics and attempted to recruit Emiko to work for them as well. She would ultimately turn on the group, however, supporting her brother Oliver in his efforts to expose the Ninth Circle and their activities to the world.
There are two chief differences between the Ninth Circle in the comics and Dante's organization on Arrow. The first is that Dante's organization is not as flamboyant or overtly evil as the Ninth Circle. The Arrow version of Dante wears designer suits rather than ceremonial robes and his organization doesn't appear require its members to undergo ritual branding or burning to show their commitment to the Ninth Circle. The other major difference is that Arrow's version isn't obsessed with having a member of the Queen family on their board of directors. In the comics, there's a prophecy that the Ninth Circle couldn't prosper without a Queen at the center of the Circle. This was why they were obsessed with recruiting Emiko to join them after it became clear Oliver Queen would never be persuaded to join them.
The Ninth Circle In Arrowverse's Future
The Ninth Circle's prophecy also plays into one theory regarding the alternate future of Star City 2038 and why Felicity Smoak felt the need to hide her pregnancy and the existence of her daughter from all but her closest friends. If Dante's organization has a similar prophecy in the Arrowverse, they would stop at nothing to acquire the last legitimate child of the Queen dynasty and might force the already paranoid Felicity into hiding. This would also explain why the Ninth Circle might ignore Thea Queen (who wasn't born of Robert Queen's lineage) and William Clayton, who was born out of wedlock.
Ignoring that theory, there's still reason to believe that the Ninth Circle is involved in the future of Star City. The plot of Arrow's flash-forward storylines has seen the vigilantes of Star City working to stop a bombing plot that would destroy all of the slums of Star City while leaving the Glades - the ritzy gated community now sitting in the center of the city - completely untouched. We also know that Rene Ramirez, now mayor of the Glades, is aware of the bombing plot, but that his silence has been bought by a mysterious figure who's involved in financing Rene's bid for reelection.
Manipulating politicians is definitely a ploy utilized by Dante's organization in the modern day Arrowverse, but there's another connection here that further ties this future group to the Ninth Circle of the Green Arrow comics. One of the Ninth Circle's schemes involved a plot to destroy Green Arrow's hometown of Seattle with a series of controlled disasters, allowing the local industry and politicians to come in and create a new city built upon the ideals of laissez-faire capitalism, with no taxes, a weak local government, and no social services beyond what the businesses financed. It remains to be seen if this is the ultimate endgame of the bombers in the future of Arrow, but it does seem like a distinct possibility.