A low-complexity adaptive equalizer including equalization, rotation of state of polarization (RSOP) tracking, and phase recovery is proposed for coherent optical polarization-division-multiplexing (PDM) quadrature phase-shifting keying (QPSK) transmission system. A conventional N-tap 2x2 butterfly finite impulse response filter is simplified to two N-tap filters and a 1-tap 2x2 filter. Two N-tap filters could compensate for inter-symbol interference and residual chromatic dispersion based on the same error cost function from the power sum of two polarization coupling constant modulus signals. 1-tap 2x2 filter which works on the nonlinear principal component analysis (NPCA) criterion is designed for joint polarization demultiplexing and phase recovery. NPCA is proved to not only void the singularity problem in the constant modulus algorithm (CMA) but also has faster tracking capability. Two kinds of adaptive equalizers using least mean squares NPCA (LMS-NPCA) and recursive least squares NPCA (RLS-NPCA) are compared in our simulation and experiment. In an experiment of 112 Gb/s PDM-QPSK transmission over 50 km standard single-mode fiber, the proposed two schemes have better transmission performance at RSOP speed of 1 Mrad/s and 5 Mrad/s compared to the conventional CMA and Viterbi-Viterbi phase estimation (VVPE) scheme. The reduced complexity of the proposed schemes is more than 40%.