The genetic basis for the adaptive advantages of polyploids over their diploid relatives remains poorly understood. To address this knowledge gap, we generate a haplotype-resolved autotetraploid alfalfa (Medicago sativa subsp. sativa) genome and construct a super-pangenome from 13 genomes across seven Medicago taxa. We discover substantial gene content variation in alfalfa, with only 20.1% of genes present on all four haplotypes. Within this group, 53.3% are core genes conserved across the Medicago genus, which we term 'tetra-copy core genes'. We find these genes are significantly enriched in climate-adaptation-associated genes (1.60-fold) and stress-responsive differentially expressed genes (1.61-fold). Paradoxically, they also carry a high genetic burden, with 80.1% of deleterious variants located in coding regions. Indeed, overexpressing a representative tetra-copy core gene, the glycine decarboxylase (MsGDC), improves both biomass and nitrogen use efficiency, despite its high genetic burden. Our study reveals the trade-off between adaptation and evolutionary constraints mediated by tetra-copy core genes, facilitating polyploid genetics and alfalfa breeding.