Wales and New Zealand, both unbeaten in their respective pools, will play runners-up Canada and Argentina respectively, while England play Pool C winners Fiji and South Africa take on the United States in the quarter-final games.
Scotland, who lost to England and New Zealand and only beat Portugal 26-5 in their group games, will play Spain in the Bowl quarter-finals.
Also competing in the Bowl competition after finishing third and fourth in their pools are Samoa, who play Russia, Australia who face Portugal and France who take on Kenya.
New Zealand will be the hot favourites to claim the title in the eighth round of the HSBC Sevens World Series after they beat England 31-5 and claimed whitewash victories of 40-0 over Portugal and 28-0 against Scotland.
Wales edged out Russia 28-22 and the United States 26-21 after an opening 22-14 win over France.
It was not to be Scotland's day as they tumbled into the relegation play-offs, with the win over Portugal their only success of the day.
A 29-7 defeat by England and 28-0 loss to New Zealand meant the USA duly climbed above them in the rankings by reaching the last eight.
There was more success for England, with wing Dan Norton reaching a landmark with three tries, taking him to 40 in the season - a record for an England player in a World Series.
The results mean Scotland, Portugal and Spain will now be defending their places in the World Series at Twickenham against the likes of Hong Kong, Russia and Tonga in an eight-team competition.
- Sports & Recreation