Holographic displays hold the promise of providing authentic depth cues, resulting in enhanced immersive visual experiences for near-eye applications. However, current holographic displays are hindered by speckle noise, which limits accurate reproduction of color and texture in displayed images. We present HoloChrome, a polychromatic holographic display framework designed to mitigate these limitations. HoloChrome utilizes an ultrafast, wavelength-adjustable laser and a dual-Spatial Light Modulator (SLM) architecture, enabling the multiplexing of a large set of discrete wavelengths across the visible spectrum. By leveraging spatial separation in our dual-SLM setup, we independently manipulate speckle patterns across multiple wavelengths. This novel approach effectively reduces speckle noise through incoherent averaging achieved by wavelength multiplexing. Our method is complementary to existing speckle reduction techniques, offering a new pathway to address this challenge. Furthermore, the use of polychromatic illumination broadens the achievable color gamut compared to traditional three-color primary holographic displays. Our simulations and tabletop experiments validate that HoloChrome significantly reduces speckle noise and expands the color gamut. These advancements enhance the performance of holographic near-eye displays, moving us closer to practical, immersive next-generation visual experiences.