We use a replica approach to investigate the thermodynamic properties of the random heteropolymers with persistent power-law correlations in monomer sequence. We show that this type of sequences possess proteinlike properties. In particular, we show that they can fold into stable unique three-dimensional structure (the "native" structure, in protein terminology) through two different types of pathways. One is a fast folding pathway and leads directly to the native structure. Another one, a more slower pathway, passes through the microphase separated (MPS) state and includes a number of intermediate glassy states. The scale and the magnitude of the MPS are calculated. The frozen state can be reached only by sequences with weak long-range correlations. The critical value for the correlation exponent is found, above which (strong correlations) freezing is impossible.