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Sharing genome data in a privacy-preserving way stands as a major bottleneck in front of the scientific progress promised by the big data era in genomics. A community-driven protocol named genomic data-sharing beacon protocol has been widely adopted for sharing genomic data. The system aims to provide a secure, easy to implement, and standardized interface for data sharing by only allowing yes/no queries on the presence of specific alleles in the dataset. However, beacon protocol was recently shown to be vulnerable against membership inference attacks. In this paper, we show that privacy threats against genomic data sharing beacons are not limited to membership inference. We identify and analyze a novel vulnerability of genomic data-sharing beacons: genome reconstruction. We show that it is possible to successfully reconstruct a substantial part of the genome of a victim when the attacker knows the victim has been added to the beacon in a recent update. We also show that even if multiple individuals are added to the beacon during the same update, it is possible to identify the victim's genome with high confidence using traits that are easily accessible by the attacker (e.g., eye and hair color). Moreover, we show how the reconstructed genome using a beacon that is not associated with a sensitive phenotype can be used for membership inference attacks to beacons with sensitive phenotypes (i.e., HIV+). The outcome of this work will guide beacon operators on when and how to update the content of the beacon. Thus, this work will be an important attempt at helping beacon operators and participants make informed decisions.