Quaker and Winged Foot (i'll assume West) are both fine examples of Tillinghast's work, but they are very different tests of golf. This is not intended as a knock.
WF is, ahem, a "Man sized" course. Long, demanding hybrid or long iron approaches (see #17, #13). With typical member rough its really hard. The green surrounds are some of my favorite,but they can be very hard. See #10 or #18.
I dont think anybody who hasn't got some kind of tour card or qualified for a USGA championship should be playing the Toonamint tees at WF, or would have fun doing so.
Saying that, I personally think WF is penal (a style, not a criticism) from the blues (blacks at WF) or whites. For example, the par threes yardage doesn't change much. #3 is 185 from the whites, 200 from the blues. #13 is 204 from the whites, 209 from the blues. at #10 the yardage changes by 4 yards. Yardage at #17 changes by 6 yards. #!8 is 409 from the whites, 430 from the blues. Thats a two and 1/2 club difference for me, but even from the whites, i'm hitting 5 or 6 iron to a green on a plateau, which has several sections demanding incredible accuracy.
QR is shorter. It demands good mid and short iron play, but precise short iron shots, and offers. to me anyway, a little more variety in terms of the sort of recovery shots demanded.
QR#11 should be on anyone's list of faves. its a great hole. The only knock I hear from people is the OB right on much of the front nine, but i've never really felt that was fair. I play away from the OB in only one place.
I like 'em both, but QR is fun course to play, and examination of the total game, but primarily the short game. WF is more of a classic Joe Dey/Nicklaus/Hogan examination. Who can hit long irons high enough and accurately enough, who can place their driver on the correct side of the fairway, etc.
i think Pat is spot on in terms of conditioning. QR has lost their greens twice that I can remember during the past five years, and i may have forgotten an episode, or more. Foliage encroachment, resulting in anthracnose, poor root structure---a lot of poor maintenance habits over a long period have come home to roost there.
People talk abt Shinny/National as a great 36 holes in a day experience, but architecturally i think of QR/WF West as a great compare/contrast.