A few things to look for as the quarterfinals get underway tomorrow:
Success Thusfar Does Not Guarantee Success Going Forward
Three of the remaining 8 teams are being written off in varying degrees because they have not been viewed as impressive to this point as some other teams: Paraguay, Uruguay, and Ghana. This does not mean that they are incapable of getting hot and marching all the way to the final. As I mentioned earlier, in the 1982 World Cup Italy started out playing poorly. They drew all 3 group stage matches and just snuck into the next round on the 2nd tiebreaker of goals scored, where Paolo Rossi caught fire and they dispatched Brasil, Argentina, Poland, and West Germany to become improbable champions. In 2002 a not-all-that-impressive Germany team rode a hot goalkeeper, some timely goals and a few other breaks all the way to the final, and Italy in 2006 barely escaped Australia in the round of 16 and (per usual) didn't really hit their stride until the quarterfinals. So to me it seems foolish to write off any team's chances at this point. We are definitely at the anything can happen stage.
Fitness and Depth
Each of these teams will be playing their 5th match in the last 16-21 days, and some are playing their quarterfinal on 3 days rest while others are on 5 days rest. This is definitely more grueling than the regular club season, where usually they play 1 match every weekend with a midweek match thrown in on many weeks. So fitness at this point is paramount. In addition to this, having a deep squad is also a massive advantage at this point(Spain, Argentina, Brasil). Lastly, all of the teams are so tired at this point that having a player sent off and having to play a long time with only 10 men will be even more of a disadvantage that it was in the first stage.
Oh, and all of this is on top of the fact that the club soccer season runs from early August to mid-May every year. In a non WC year these guys play a 10 month season and have just under 2 months off. Compare this to the 6 month offseason in futbol Americano or 4-5 month offseason in the NBA and NHL. This year they had 3-4 weeks "off" between the end of the club season and the start of the World Cup. However, the fact that most of that was spent training & preparing with their national teams means that many players have done all this in the last month on about 1 week off. Since last July.
As important as they have been to this point, from here on out they are massive. Each team has a few players that are "on a yellow", meaning that one more will means they miss the following match. Missing a key player is never good; missing one for a World Cup semifinal or final can be devastating (Ballack, Germany, World Cup 2002 final; Nedved, Czech Republic, Euro 2004 semifinal, etc). Pay attention to which players are on a yellow already and watch to see if they play any differently overall or if they avoid hard tackles specifically.