Multiplexed genetic perturbations are critical for testing functional interactions among coding or non-coding genetic elements. Compared to DNA cutting, repressive chromatin formation using CRISPR interference (CRISPRi) avoids genotoxicity and is more effective for perturbing non-coding regulatory elements in pooled assays. However, current CRISPRi pooled screening approaches are generally limited to targeting 1-3 genomic sites per cell. To develop a tool for higher-order (≥3) combinatorial targeting of genomic sites with CRISPRi in functional genomics screens, we engineered an Acidaminococcus Cas12a variant -- referred to as mul tiplexed transcriptional interference AsCas12a (multiAsCas12a). multiAsCas12a significantly outperforms state-of-the-art Cas12a variants in combinatorial CRISPRi targeting using high-order multiplexed arrays of CRISPR RNAs (crRNA) delivered by lentiviral transduction, including in high-throughput pooled screens using 6-plex crRNA array libraries. Using multiAsCas12a CRISPRi, we discover new enhancer elements and dissect the combinatorial function of cis-regulatory elements. These results demonstrate that multiAsCas12a enables group testing strategies to efficiently survey potentially numerous combinations of chromatin perturbations for biological discovery and engineering.