The Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory was designed to study the properties of quantum chromodynamics (QCD) in a hot and dense medium. The first years of RHIC operation and accompanying theoretical studies have helped pinpoint certain classes of measurements needed to more fully probe the medium and determine its properties. The medium created in these heavy-ion (AA) collisions appears to thermalize quickly and exhibits collective flow patterns consistent with hydrodynamic predictions. The initial temperature of the medium is not known and it is not yet understood whether deconfinement and chiral symmetry restoration are realized during its evolution. The answers to these questions require higher luminosities and detector upgrades, referred to as RHIC-II. The goal of RHIC II is to achieve the answers to the above questions by increasing the ion luminosity. The measurements thus far at RHIC could not fully address these fundamental questions, either due to incomplete detection capabilities or insufficient statistics to draw meaningful and robust conclusions. Working groups were formed to determine which physics topics could best be addressed by the combination of planned upgrades and increased luminosity. Reports from each working group were used to prepare a white paper for RHIC II, along with additional inputs from the conveners of all working groups.