I wonder if Quantum was a kind of "respectable" cover for SPECTRE - peopled as it was at the top by billionaires, ex-intelligence types, political advisers - which outlived its usefulness when 007 embarrassed its membership in the opera house. These people panicked and neither they nor anyone else associated with Quantum was of any further use to SPECTRE. Dominic Greene set the seal on it, and SPECTRE knew it - motor oil in his stomach certainly, but two bullets in his skull.
Quantum seemed based around "financial" activity - the whole Le Chiffre business, and then in QoS the attempt to control, legally, the world's water supplies. It might have been like a hedge fund, with SPECTRE as the main, if very secret, investor.
As for the man charged with winding Quantum up - Mr White, probably SPECTRE's enforcer within Quantum, rather than the leader of it, in my view. Which makes his apparent fall from grace glimpsed in the Spectre teaser trailer interesting - he looks a dishevelled shadow of his former self. I wonder what he's done to deserve this? Perhaps having wound up Quantum White is himself of no further use to SPECTRE? In which case, in the manner of the classic Connery films, surely SPECTRE would have had him eliminated without delay?
That's one idea I'd offer about Quantum, but as for Mr White - intriguing stuff, and as ever, we won't know for sure until November.