The long spreading code and the binary offset carrier (BOC) modulation technique are used in the next-generation Global Navigation Satellite System (GNSS) to improve positioning performance and to reduce inter-GNSS interference; however, the signal acquisition process of a GNSS receiver can take more time and requires additional hardware resources than legacy Global Positioning System (GPS) receivers. This paper presents the BOC 2-D compressed correlator (TDCC) technique for the fast acquisition of the next-generation GNSS signals. In the proposed BOC-TDCC, signal power in neighboring code-phase hypotheses and Doppler frequency hypotheses can be coherently combined and tested by a single compressed correlator in the first stage, and the conventional correlator-based serial search technique is employed in the second stage to test each hypothesis combined in the first stage. The performance of the BOC-TDCC is demonstrated with numerous Monte Carlo simulations and tested with real data. The BOC-TDCC has much lower mean acquisition time (MAT) than other conventional search schemes, which demonstrates that the BOC-TDCC is an effective search scheme for next-generation GNSS signals.