Harlequins have scored 27 tries in their five winning group matches, with Tom Williams collecting a brace against a game but outgunned Connacht side as Conor O'Shea's men marched into the knockout stages for only the second time since 1998.
Toulon are also in the knockout stages after beating Cardiff Blues 45-25 at Stade Felix Mayol, with the try bonus point taking them to 23 points.
They still have to go to Montpellier - who are on 18 points - in Round 6 but Toulon are assured of being in the mix in April as group winners at best or one of the two best runners-up at worst.
And they are through to successive quarter-finals, having made the final eight in their most recent appearance in the tournament two seasons ago.
Ulster Rugby were the first last-eight qualifiers when they beat Glasgow Warriors 23-6 under the Ravenhill floodlights.
It took them five points clear of Northampton Saints in Pool 4 and through to the knockout stages with a match to spare courtesy of their superior 6-4 match points advantage over the Saints from their head-to-head encounters.
And, just like Toulon, they still have plenty to play for in respect of winning priceless home advantage when they go to Castres Olympique next week.