Sleep　staging　is　essential　in　assessing　sleep　quality　and　diagnosing　sleep-related　disorders,　but　the　lack　of　labeled　data　impedes　the　development　of　automatic　sleep　staging　models.　Generally,　institutions　rely　on　semi-supervised　approaches　to　enhance　the　utilization　of　their　own　unlabeled　data.　However,　the　task　knowledge　obtained　from　a　limited　amount　of　labeled　data　is　often　insufficient　to　guide　the　learning　based　on　large　amounts　of　unlabeled　data,　which　may　even　lead　to　catastrophic　forgetting　and　further　degrade　the　performance　of　most　existing　methods.　In　this　paper,　we　propose　a　novel　strategy　of　building　secure　collaboration　among　multiple　institutions,　to　achieve　the　implicit　augmentation　of　labeled　data　and　expansion　of　task　knowledge　for　each　participating　institution　by　acquiring　external　knowledge　from　others.　We　adopt　the　Federated　Learning　(FL)　to　facilitate　secure　collaboration　and　propose　a　federated　semi-supervised　sleep　staging　method　based　on　knowledge　sharing,　which　enables　the　automatic　scoring　of　sleep　stages　using　only　single-channel　EEG　data.　The　task　knowledge　in　our　method　is　contained　in　relationships,　which　exist　naturally　among　sleep　stages　and　can　be　extracted　from　both　local　labeled　and　unlabeled　data.　Furthermore,　the　knowledge　sharing　among　participating　institutions　can　be　achieved　by　aligning　the　local　relationships　to　the　aggregated　global　relationships.　Additionally,　we　employ　prototype-contrastive　learning　to　enhance　the　clarity　of　relationships　extracted　from　labeled　data,　and　propose　pseudo-labeling　optimization　to　generate　reliable　pseudo-labels　for　subsequent　relationship　extraction　from　unlabeled　data.　Our　method　is　shown　to　be　effective　and　outperforms　compared　methods　in　extensive　experiments　conducted　on　two　publicly　available　datasets.　©　2023　Elsevier　Ltd