We study mid-infrared quantum-cascade lasers (QCL's) with a pair of triply harmonic resonant levels. Potential applications of such designs are discussed based on the resonant third-order nonlinear susceptibility χ(3) at the third-harmonic (TH) frequency, χ(3)(3ω; ω, ω, ω), and that at fundamental mode (FM) frequency χ(3)(ω; ω, ω,-ω). Mode overlap and the phase mismatch effect are considered in the TH power evaluation. In addition to single-photon processes, resonant two-photon processes are included in the evaluation of χ(3)(ω; ω, ω,-ω), which results in the enhancement of the positive nonlinear (Kerr) refractive index, and thus induce stronger self-phase modulation (SPM). It is concluded that QCL's with multiple-resonance design are not only favorable for tunable light emission in the near- to mid-infrared region, but are also promising candidates for spectrum broadening by SPM.