We review recent studies on developing tools for quantum complex phenomena. The tools have been applied for clarifying the perspective of the Mott transitions and the phase diagram of metals, Mott insulators and magnetically ordered phases in the two‐dimensional Hubbard model. The path‐integral renormalization‐group (PIRG) method has made it possible to numerically study correlated electrons even with geometrical frustration effects without biases . It has numerically clarified the phase diagram at zero temperature,T= 0, in the parameter space of the onsite Coulomb repulsion, the geometrical frustration amplitude and the chemical potential. When the bandwidth is controlled at half filling, the first‐order transition between insulating and metallic phases is evidenced. In contrast, the filling‐control transition shows diverging critical fluctuations for spin and charge responses with decreasing doping concentration. Near the Mott transition, a nonmagnetic spin‐liquid phase appears in a region with large frustration effects. The phase is characterized remarkably by gapless spin excitations and the vanishing dispersion of spin excitations. Magnetic orders quantum mechanically melt through diverging magnon mass. The correlator projection method (CPM) is formulated as an extension of the operator projection theory. This method also allows an extension of the dynamical mean‐field theory (DMFT) with systematic inclusion of the momentum dependence in the self‐energy. It has enabled determining the phase diagram atT> 0, where the boundary surface of the first‐order metal‐insulator transition at half filling terminates on the critical end curve atT=Tc. The critical end curve is characterized by the diverging compressibility. The single particle spectra show strong renormalization of low‐energy spectra, generating largely momentum dependent and flat dispersion. The results of two tools consistently suggest that the strong competitions of various phases with underlying diverging compressibility can induce an emergence near a new type of quantum criticality distinguished from the conventional and simple quantum critical phenomena. © 2003 American Institute of Physics