But recent tournament history underlines it is not a trip Aviva Premiership champions Quins can take lightly.
Their quarter-final hopes last season were destroyed by a 9-8 loss to Connacht on the final weekend of pool stage games. Had they won, Quins would have progressed.
"We know that we face a massive challenge when we go to Galway," Quins rugby director Conor O'Shea said. "Last season is etched in our memory and we are looking forward to getting over there.
"I know the way (Connacht coach) Eric Elwood is. He has a really ambitious way of playing now, and people write them up in a different way to what they should.
"They have a huge number of youngsters and they have a really, really physical pack. We know what their season is based around."
Saracens, fresh from a 45-0 drubbing of Edinburgh in Scotland last weekend, continue on their Pool One travels tomorrow when they host Paris-based Racing Metro at Brussels' King Baudouin Stadium.
Racing, meanwhile, have announced that January's return fixture between the clubs will take place in Nantes.
Exeter, who produced one of the opening round's outstanding performances in losing just 9-6 to holders Leinster on their Heineken Cup debut, face another giant task tomorrow.
French heavyweights Clermont Auvergne, 49-16 conquerors of the Scarlets six days ago, are the Pool Five visitors to Sandy Park for what could prove a memorable encounter.
And the Scarlets, needing to get back on track at home to Leinster tomorrow, have been boosted by Wales international Morgan Stoddart's availability.
Stoddart, sent off against Clermont, is free to continue playing after tournament disciplinary chiefs decided a red card was sufficient punishment.
Northampton will need to overcome a five-day turnaround when they resume European business in France tonight.
Saints head to the Stade Ernest Wallon in Toulouse, where Pool Four rivals Castres await them.
While Castres have had a week to recover and prepare after being beaten by Ulster, Saints were in European action against Glasgow last Sunday.
"The tournament scheduling has not been particularly kind to us in the first two rounds," Northampton rugby director Jim Mallinder said. "And having just five days to prepare for a tough match is certainly not ideal.
"However, we are going to France in good spirits after showing our character last Sunday to fight back and secure a bonus-point win, and we will be doing everything we can to get the positive result we want."
In the second-tier Amlin Challenge Cup tomorrow, London Irish and Worcester will look to build on winning starts when they host Mont-de-Marsan and Rovigo respectively, while London Welsh visit Italian side Prato.
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