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Operation	Hyperproofs cost	Verkle trees cost (KZG, arity = B)
Commit	n 𝔾1	~ n 𝔾1
Compute all proofs	n log(n) 𝔾1	~n lgB
Update digest	1 𝔾1	$\log_B(n)$ 𝔾1
Update all proofs (”state”)	log n 𝔾1	$\log_B(n)$ 𝔾1
Verify 1 proof	log n ℙ	$\log_B(n)$ ℙ
Aggregate m proofs	m log n (𝔾1+𝔾2+ℙ)
Verify aggr. of m proofs	m log n (𝔾2+𝔾T+ℙ)	no aggregation
Size of one proof	log n 𝔾1	$(\log_B(n)$ +1 ) 𝔾1
Size of aggregated proof	log(m log n) 𝔾T	no aggregation
Group requirements
Aggregation	via IPA (costly)	no aggregation
Updates	Yes	No updates
Maintainability (update all)	Yes	Yes
Compute/Update a fraction ⁍ of the tree	$O(\alpha)$	$O(n \log B)$
Total # of proofs in Verkle Trees	NA	$B^d + B^{d-1} + ... + B$ 𝔾1
Notes	NA	d is depth of VT; B is branching factor. It holds that $n = B^d$ , i.e., $d = \log_B(n)$