An alternating ring-opening copolymerization (ROCOP) of epoxides (cyclohexene oxide, limonene oxide) and anhydrides (phthalic anhydride, naphthalic anhydride and maleic anhydride) was performed using a chromium complex containing a sterically demanding amino triphenolate ligand in the presence of a cocatalyst (bis(triphenylphosphine)iminium chloride (PPNCl) or 4-dimethylaminopyridine (DMAP)). The polymerization reaction was strongly affected by the types and feed ratio of monomers and cocatalysts. After optimizing the reaction conditions, the chromium complex successfully produced perfectly alternating polyesters with high selectivity (>99% ester linkages), high molecular weights (up to 43.8 kDa), high glass transition temperatures (T_g up to 159 °C), and low dispersities. In addition, a small amount of the catalyst yielded a turnover frequency (TOF) as high as 380 h⁻¹.