(covers information from several alternate timelines)
The quantum level was a term used for activity that took place on a sub-atomic scale.
In 2370, in a quantum reality where Captain Picard was killed by the Borg, Commander Data explained to Lieutenant Worf and Captain Riker the properties of the quantum level, when it was discovered that Worf was moving through different quantum realities. (TNG: "Parallels")
When Jadzia Dax and Deral were working on a way to keep the planet Meridian longer in the current dimension they discovered that the fusion reactants in the core were out of balance. This imbalance was building toward a cascade reaction on the quantum level. (DS9: "Meridian")
When the senior staff of Deep Space 9 was trapped in a holographic program in a holosuite, Rom told Michael Eddington that neural energy has to be stored on a quantum level. The holosuite couldn't handle that. (DS9: "Our Man Bashir")