This paper proposes a new family of approximate QR-based least-squares (LS) adaptive algorithms called p-A-QR-LS for blind minimum output energy (MOE) detection in CDMA communications systems. It extends the A-QR-LS algorithm by retaining different numbers of diagonal plus p - 1 off-diagonals of the triangular factor of the augmented data matrix. For p = 1 and N (N is the length of the weighting vector), it reduces to the A-QR-LS and the QR-RLS algorithms, respectively. It not only provides a link between the QR-LMS-type and the QR-RLS algorithms through a well-structured family of algorithms, but also offers more freedom in the complexity-performance tradeoffs for practical receiver design in communication systems. The performance of the proposed algorithm is verified by computer simulations.