How many pilots do long flights need?
The regulations for this are in 14 CFR 121 Subpart R—FLIGHT TIME LIMITATIONS: FLAG OPERATIONS.

In short, flight time limits within a rolling 24 hour period depend on the number of crew members assigned to the flight:


One or two pilot crews - 8 hours max
Two pilots and one additional flight crewmember: 12 hours max
Three or more pilots and an additional flight crewmember: More than 12 hours, but "adequate sleeping quarters on the airplane" will be provided


However, they may also implement a "Fatigue Risk Management System" which allows them to exceed these limits if they are found to be safe and approved by the FAA.  This system would most likely address the number of timezones crossed but is optional.

A particular airlines pilot contract may also have more restrictive limitations (like considering the number of timezones crossed) that are negotiated by the union, but this is not required by the regulations.  




  §121.481   Flight time limitations: One or two pilot crews.  
  
  (a) A certificate holder conducting flag operations may schedule a
  pilot to fly in an airplane that has a crew of one or two pilots for
  eight hours or less during any 24 consecutive hours without a rest
  period during these eight hours.
  
  ...
  
  §121.483   Flight time limitations: Two pilots and one additional flight crewmember.  
  
  (a) No certificate holder conducting flag operations may schedule a
  pilot to fly, in an airplane that has a crew of two pilots and at
  least one additional flight crewmember, for a total of more than 12
  hours during any 24 consecutive hours.
  
  ...
  
  §121.485   Flight time limitations: Three or more pilots and an additional flight crewmember.  
  
  (a) Each certificate holder conducting flag operations shall schedule
  its flight hours to provide adequate rest periods on the ground for
  each pilot who is away from his base and who is a pilot on an airplane
  that has a crew of three or more pilots and an additional flight
  crewmember. It shall also provide adequate sleeping quarters on the
  airplane whenever a pilot is scheduled to fly more than 12 hours
  during any 24 consecutive hours.
  
  ...
  
  §121.495   Fatigue risk management system.  
  
  (a) No certificate holder may exceed any provision of this subpart
  unless approved by the FAA under a Fatigue Risk Management System.
  
  (b) The Fatigue Risk Management System must include:
  
  (1) A fatigue risk management policy.
  
  (2) An education and awareness training program.
  
  (3) A fatigue reporting system.
  
  (4) A system for monitoring flightcrew fatigue.
  
  (5) An incident reporting process.
  
  (6) A performance evaluation.