[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85161-en":3,"doc-seo-85161-105":29,"detail-sidebar-cat-0-en-105":83},{"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},85161,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries","Near-maximum Boolean circuit lower bounds are established for the complexity class EprMA/1, which corresponds to exponential time with access to a promise-MA oracle plus one bit of advice. The argument uses the iterative win-win paradigm, a reduction from the Range Avoidance problem to circuit lower bounds, and the PCP theorem. Central analysis studies PNP with adaptive NP query rounds and bounded witness length, quantified via an auxiliary complexity measure.","Near-Maximum Circuit Lower Bounds for Exponential Time  \nwith Merlin-Arthur Queries  \nHanlin Ren* IAS  \n[h4n1in.r3n@gmail.com](h4n1in.r3n@gmail.com)  \nRyan Williams† MIT  \n[rrw@mit.edu](rrw@mit.edu)  \narXiv :2607 .09963v 1 [ cs .CC] 10 Jul 2026  \nJuly 14, 2026  \nAbstract  \nWe prove a near-maximum (2n /n) circuit lower bound for the complexity class EprMA/ 1, corresponding to exponential time with access to a promise-MA oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen–Lu–Oliveira–Ren–Santhanam, FOCS’23), the reduction from the Range Avoidance problem to circuit lower bounds (Jeˇrábek, Ann. Pure Appl. Log.’04; Korten, FOCS’21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class  \nPNP 􀀔\\#roleunndgtsh  \n=  \n=  \nrs􀀕 , which is PNP with r(n) adaptive rounds of NP queries, where each NP query has  \nwitness length s(n) .  \nContents  \n1 Introduction 1  \n1.1 Our Results ............................................. 1  \n1.2 Techniques .............................................. 3  \n1.2.1 The Iterative Win-Win Method .............................. 3  \n1.2.2 Towards Solving Range Avoidance ............................ 4  \n2 Preliminaries 5  \n2.1 Smart Reductions to Promise Problems ............................. 6  \n2.2 Merlin-Arthur Classes ....................................... 6  \n2.3 Self-Corrector for Low-Degree Polynomials ........................... 7  \n3 Bounded-Adaptive Queries to an NP Oracle 7  \n3.1 The Jeˇrábek–Korten Reduction .................................. 8  \n3.2 Encoded Computational History ................................. 9  \n4 A Near-Maximum Circuit Lower Bound 13  \n4.1 Instance-Wise Hardness-Randomness Tradeoff for AVOID ................... 14  \n4.2 The Lower Bound .......................................... 16  \n5 Discussion 19  \nReferences 21  \nA Half-Exponential Circuit Lower Bounds for EprMA 24  \n* Hanlin Ren is supported by the Massive Dynamics Member Fund at the Institute for Advanced Study.  \n†This work was initiated while visiting the Institute for Advanced Study, Princeton, NJ. This material is based upon work supported by the National Science Foundation under grants DMS-2424441 (at IAS) and CCF-2420092 (at MIT) .  \n1 Introduction  \nA simple counting argument dating back to Shannon [Sha49] shows that a uniformly random 2nbit truth table describes a near-maximum hard function requiring Boolean circuits of size Ω(2n /n) on inputs of length n, with high probability: there are simply too many possible functions to be covered by all circuits of size 2n /(10n) . Significant research has gone into understanding how efficiently such a hard function can be constructed, and this problem is central to complexity theory. For example, if there are functions in NP with near-maximum hard (or even superpolynomially hard) finite slices, then P  NP. If there are near-maximum hard (or even exponentially-hard) functions in E = TIME[2O (n)], then P = BPP [IW97] . However, the non-uniformity of circuit families makes the uniform construction of hard functions apparently extremely difficult. It has been known for decades that the (huge) class Σ3 E = Σ3 TIME[2O (n)] contains a function of near-maximum circuit size [Kan82] and that the complexity can be reduced slightly to EΣ2 P [MVW99] .  \nIn the last few years, substantial progress has been made on understanding the uniform complexity of near-maximum hard functions. Chen, Hirahara, and Ren [CHR24] showed that Σ2 E (indeed, even S2 E/1 ) contains near-maximum hard functions, resolving a 40-year open problem, using a novel winwin paradigm for constructing hard functions. Li [Li24] gave a drastically simplified proof that also improved the lower bound in several aspects; see [CHLR26] for a combined exposition. More recently, Chen, Li, and Liang [CLL25] showed that exponential time Arthur-Merlin with subexponential advice (AMEXP/2nε) contains near-maximum hard functions.  \nThese recent results raise the questi","cbCair0xShldrQY1","https://ap.wps.com/l/cbCair0xShldrQY1","pdf",446439,1,25,"English","en",105,"# Introduction\n## Our Results\n## Techniques\n## The Iterative Win-Win Method\n## Towards Solving Range Avoidance\n# Preliminaries\n## Smart Reductions to Promise Problems\n## Merlin-Arthur Classes\n## Self-Corrector for Low-Degree Polynomials\n# Bounded-Adaptive Queries to an NP Oracle\n## The Ježábek–Korten Reduction\n## Encoded Computational History\n# A Near-Maximum Circuit Lower Bound\n## Instance-Wise Hardness-Randomness Tradeoff for AVOID\n## The Lower Bound\n# Discussion\n# References","[{\"question\":\"What role does the analysis of PNP with adaptive NP queries play?\",\"answer\":\"It characterizes PNP with r(n) adaptive NP query rounds and bounded witness length s(n), which is crucial for deriving the final near-maximum circuit lower bound for the target class.\"}]",1784201464,63,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"near-maximum-circuit-lower-bounds-for-exponential-time-with-merlin-arthur-queries","",{"@graph":35,"@context":77},[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/near-maximum-circuit-lower-bounds-for-exponential-time-with-merlin-arthur-queries/85161/",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],{"name":72,"@type":73,"acceptedAnswer":74},"What role does the analysis of PNP with adaptive NP queries play?","Question",{"text":75,"@type":76},"It characterizes PNP with r(n) adaptive NP query rounds and bounded witness length s(n), which is crucial for deriving the final near-maximum circuit lower bound for the target class.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]