[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82869-en":3,"doc-seo-82869-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82869,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Symmetry all the way down","Asymmetric trust generalizes classical symmetric quorum systems by letting each process assume its own failure scenarios. This added flexibility raises the question of whether distributed tasks become solvable in stronger cases, and with what limits. Using the depth hierarchy for asymmetric trust, the work proves that asymmetric trust does not increase solvability for tasks requiring depth two or more, such as reliable broadcast and consensus. Any Byzantine failure scenario enabling depth≥2 solutions can be replicated by a suitably constructed Byzantine symmetric quorum system via a compiler, and the remaining advantages apply only to depth≤1 tasks. ","arXiv :2607 .04887v 1 [ cs .DC] 6 Jul 2026  \nSymmetry all the way down  \nIgnacio Amores-Sesar 1 , Christian Cachin2 , Simon Holmgaard Kamp3 , and Juan Villacis2  \n1 Aarhus University, Denmark  \n2 University of Bern, Switzerland  \n2 Ruhr University Bochum, Germany  \nAbstract  \nAsymmetric trust generalizes classical symmetric quorum systems by allowing each process to specify its own failure assumptions. While this flexibility enables tolerance of strictly more failure scenarios, it is not known if, in these cases, it is actually possible to solve distributed tasks, and if so, which.  \nWe answer this question using the depth hierarchy for asymmetric trust (Amores-Sesar et al., OPODIS ’25), which characterizes how much a process must rely on others to solve a task. We prove that asymmetric trust does not increase the solvability of tasks requiring depth two or more, such as reliable broadcast or consensus. Specifically, for any Byzantine asymmetric quorum system, every failure scenario that permits solving a task requiring depth at least two can also be tolerated by a suitably constructed Byzantine symmetric quorum system. We show this via a compiler that transforms asymmetric quorum systems into symmetric ones.  \nThe additional failure patterns tolerated exclusively by asymmetric trust correspond to scenarios in which only simpler tasks requiring depth one or less (such as consistent broadcast) can be solved. We further prove that this result is tight in the depth hierarchy, meaning that there exist no compilers that produce symmetric quorum systems that are valid also in failure scenarios where correct processes have depths one or less.  \nOur results clarify the precise power of asymmetric trust. While it strictly enlarges theset of tolerable failure patterns, it does not provide additional strength for solving tasks requiring depth two or higher.  \n1 Introduction  \nSolving problems in distributed systems requires making explicit assumptions about the types and patterns of failures that may occur. Most classical systems rely on global adversarial thresholds, which assume that at most f out of n processes may be faulty at any given time. A more general abstraction is provided by symmetric quorum systems [25] . Instead of bounding the number of faulty processes, quorum systems characterize the sets of processes that may concurrently fail, often based on shared properties such as geographic location, operating system, or administrative domain. Under these models, all participants adopt the same trust assumptions. Accordingly, these are referred to as symmetric trust assumptions, and protocols built upon them as symmetric protocols.  \nHowever, the symmetric model does not capture the heterogeneous and subjective trust relationships that occur in the real world. To address this limitation, asymmetric trust was introduced [13] . In this setting, each process operates under its own trust assumptions, formalized through an asymmetric quorum system. Processes may define their quorums independently, reflecting individual views of which failures are expected. Protocols designed for this environment are known as asymmetric protocols.  \nAnother benefit provided by asymmetric trust is that, in fully permissionless environments [20], it provides an alternative to traditional consensus protocols [27, 29] that mitigate Sybil attacks  \nby requiring external resources, such as computational power or capital. A prominent example is the Stellar Consensus Protocol (SCP) [26], which relies exclusively on local, subjective trust decisions, from which global asymmetric quorums emerge organically. By decoupling security from physical or economic resources, asymmetric trust enables consensus in permissionless networks through a completely different approach, based solely on trust relationships, which more closely resembles traditional permissioned BFT algorithms. This gives rise to more efficient and fast protocols.  \nAsymmetric trust has been extensivel","cbCaioDg4nMoUdhA","https://ap.wps.com/l/cbCaioDg4nMoUdhA","pdf",404370,1,18,"English","en",105,"# Abstract\n# Introduction\n## Symmetric versus asymmetric trust\n## Depth hierarchy and task solvability\n## Main results and their implications","[{\"question\":\"What is asymmetric trust in quorum systems?\",\"answer\":\"Asymmetric trust allows each process to specify its own failure assumptions through its own asymmetric quorum system. Protocols built on these assumptions are called asymmetric protocols.\"},{\"question\":\"How does depth hierarchy affect what tasks are solvable?\",\"answer\":\"Depth zero processes cannot solve any tasks. Depth one supports only simple tasks such as consistent broadcast, while tasks requiring depth at least two include reliable broadcast and consensus.\"},{\"question\":\"Does asymmetric trust enable solving tasks like reliable broadcast or consensus under more failure patterns?\",\"answer\":\"No. The results show asymmetric trust does not increase solvability for tasks requiring depth two or higher. Failure scenarios that enable such tasks can be tolerated by a constructed symmetric quorum system using a compiler.\"}]",1784183569,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"symmetry-all-the-way-down","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/symmetry-all-the-way-down/82869/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is asymmetric trust in quorum systems?","Question",{"text":75,"@type":76},"Asymmetric trust allows each process to specify its own failure assumptions through its own asymmetric quorum system. Protocols built on these assumptions are called asymmetric protocols.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does depth hierarchy affect what tasks are solvable?",{"text":80,"@type":76},"Depth zero processes cannot solve any tasks. Depth one supports only simple tasks such as consistent broadcast, while tasks requiring depth at least two include reliable broadcast and consensus.",{"name":82,"@type":73,"acceptedAnswer":83},"Does asymmetric trust enable solving tasks like reliable broadcast or consensus under more failure patterns?",{"text":84,"@type":76},"No. The results show asymmetric trust does not increase solvability for tasks requiring depth two or higher. Failure scenarios that enable such tasks can be tolerated by a constructed symmetric quorum system using a compiler.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]