Bits Under ZK-LLM: Evaluating Zero-Knowledge-Friendly Quantization for Verifiable Private LLM Inference

Taeung Yoon; Yupeng Zhang; Xiaojing Liao

cs.AISep 29, 2026

Bits Under ZK-LLM: Evaluating Zero-Knowledge-Friendly Quantization for Verifiable Private LLM Inference

Authors: Taeung Yoon, Yupeng Zhang, Xiaojing Liao

Organizations: Siebel School of Computing and Data Science University of Illinois Urbana-Champaign

Abstract

Zero-knowledge proofs are emerging as a promising approach for enabling private, verifiable LLM governance and auditing, where regulators, users, and auditors need to verify claims about training-data usage or LLM inference-time behavior, while model providers must protect proprietary model parameters. However, despite the growing interest in ZK-LLMs, the understanding of ZK-friendly quantization remains limited. This gap matters because in the ZK setting, quantization directly shapes the arithmetic structure, constraint complexity, and proving cost of ZK inference. ZK protocols operate over finite fields and incur costs that depend heavily on the number and type of arithmetic operations, nonlinearities, and lookup constraints. Understanding ZK-friendly quantization is therefore essential for making ZK-LLMs practical. In this work, we present the first systematic study of ZK-friendly quantization for LLMs. We first formalize the definition of ZK-friendly quantization, capturing the properties required for ZK proof generation. We then evaluate nine language models, including Qwen2.5-14B and the mixture-of-experts model Qwen3-30B-A3B, across a broad design space of weight, activation, and nonlinear lookup table precision. Our results show that activation precision is substantially more sensitive than weight precision, while nonlinear lookup approximations can become the dominant source of utility degradation. Also, we identify RMSNorm inverse-square-root lookups as a recurring bottleneck in several large models and recover near-baseline utility by selectively increasing precision only at the bottleneck. Finally, we show that reducing bit-width or lookup-table size does not necessarily yield proportional end-to-end proving savings, showing that conventional low-bit quantization heuristics do not directly translate to ZK proving efficiency and motivating operator-aware precision selection.

Figures & tables

WikiText-2 C4 Model B8 B12 B16 B8 B12 B16 GPT-2 S 1866.2 1.011 1.000 1439.4 1.018 1.002 GPT-2 M 3213.6 1.002 1.001 1465.4 1.000 0.999 Llama 8B $>10^{4}$ 1247.9 1126.8 $>10^{4}$ 422.2 380.5 Qwen 14B $>10^{4}$ $>10^{4}$ 32.5 $>10^{4}$ $>10^{4}$ 18.3 Qwen3 30B $>10^{4}$ 148.2 131.3 $>10^{4}$ 78.7 120.7	LUT configuration Model Full B16 RMS only RMS exact RMS B24 Llama 8B 1126.8453 1126.7972 0.9993 1.0002 Qwen 7B 1496.9087 1528.4988 0.9999 0.9987 Qwen 14B 32.4609 32.6139 1.0002 0.9997 Qwen3 30B 131.3352 1.0398 1.0034 1.0014
(a) LUT precision sensitivity	(b) Targeted precision adjustment

Table 1: PPL ratios relative to the W16A16 linear-only baseline: (a) full-LUT precision and (b) operator-wise diagnosis on WikiText-2. Complete results for all models are provided in Appendix D .

Projection	Floating-point	Integer
(each)	(estimated)	(measured)
Q, O	12,158.78	11.877
K, V	2,431.76	4.603
FFN gate, up	32,828.71	20.985
FFN down	27,105.54	20.526
Sum	121,944.05	95.457

Table 2: Full-width projection proving times (seconds).

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

			Simulation error $E_{\mathrm{sim}}$
Scale	Precision	Quantization error $E_{\mathrm{quant}}$	FP32	BF16
Standard	W8A8	$6.8022\times 10^{-1}$	$2.2056\times 10^{-7}$	$2.4294\times 10^{-3}$
	W12A12	$1.2246\times 10^{-2}$	$3.1378\times 10^{-7}$	$2.2690\times 10^{-3}$
	W16A16	$7.2081\times 10^{-4}$	$3.1691\times 10^{-7}$	$1.7096\times 10^{-3}$
PoT	W8A8	$6.7154\times 10^{-1}$	$0$	$1.6644\times 10^{-3}$
	W12A12	$2.2929\times 10^{-2}$	$0$	$1.6218\times 10^{-3}$

Appendix

Table 3: Numerical validation of QDQ on actual GPT-2 Small projections. All values are relative errors, not percentages.

Dataset	$B$	GPT-2			TinyLlama	Llama-3.1	Qwen2.5			Qwen3
Dataset	$B$	Small	Medium	XL	1.1B	8B	3B	7B	14B	30B-A3B
WikiText-2	8	1866.2	3213.6	704.3	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$
	12	1.011	1.002	1.002	2356.3	1247.9	125.5	5254.5	$>10^{4}$	148.2
	16	1.000	1.001	1.000	3.073	1126.8	1.000	1496.9	32.5	131.3
C4	8	1439.4	1465.4	316.7	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$	$>10^{4}$
	12	1.018	1.000	0.989	2258.8	422.2	78.8	3472.9	$>10^{4}$	78.7

Appendix

Table 4: Full-LUT PPL ratios relative to the W16A16 linear-only baseline.

LUT configuration	Llama-3.1-8B	Qwen2.5-7B	Qwen2.5-14B	Qwen3-30B-A3B
Full B16	1126.8453	1496.9087	32.4609	131.3352
RMS B16 only; others exact	1126.7972	1528.4988	32.6139	1.0398
Full B16; RMS exact	0.9993	0.9999	1.0002	1.0034
Full B16; RMS B24	1.0002	0.9987	0.9997	1.0014

Appendix

Table 5: PPL ratios relative to the W16A16 linear-only baseline with targeted precision adjustment.

	GPT-2 Small			GPT-2 Medium			GPT-2 Large
Configuration	Mean	Min	Max	Mean	Min	Max	Mean	Min	Max
W16A16	104.474	103.257	105.087	222.355	218.421	227.839	525.555	517.006	531.283
W8A16	102.718	102.015	103.790	221.179	218.017	224.541	524.634	523.111	525.836
W10A16	103.205	102.106	103.921	221.231	216.977	225.548	526.105	519.276	535.732
W12A16	104.079	101.009	106.103	217.248	212.212	220.218	527.309	524.410	530.007
W14A16	104.683	102.837	106.115	215.175	212.812	217.813	535.667	529.128	539.949

Appendix

Table 6: Proving time under different W/A precisions (seconds).

	GPT-2 Small		GPT-2 Medium
Context length	W16A16	W12A16	W16A16	W12A16
4	73.116	74.093	158.405	150.433
8	87.156	85.499	183.162	179.619
16	103.666	100.756	225.444	218.262
32	125.902	130.727	274.920	274.536

Appendix

Table 7: Proving time across context lengths (seconds).

		WikiText-2			C4
Model	Configuration	KL $\downarrow$	Top-5 $\uparrow$	Bottom-20% $\uparrow$	KL $\downarrow$	Top-5 $\uparrow$	Bottom-20% $\uparrow$
Qwen2.5-3B	W16A16, linear	0.001847	0.9726	0.9872	0.001570	0.9769	0.9884
	W16A16, full	0.003710	0.9612	0.9835	0.002999	0.9695	0.9853
	W12A12, full	0.040696	0.8842	0.9505	0.030019	0.9032	0.9536
Qwen2.5-7B	W16A16, linear	0.015128	0.9354	0.9719	0.019180	0.9307	0.9670
	W16A16, full	0.014987	0.9363	0.9701	0.013916	0.9395	0.9744

Appendix

Table 8: Logit-level fidelity relative to the BF16 reference.

		W12A12						W16A16
Model	Dataset	Base	PoT	Conv.	$c_{1}/c_{2}$	Asym	Dyn	Base	PoT	Conv.	$c_{1}/c_{2}$	Asym	Dyn
GPT-2 Small	WikiText-2	27.05	28.53	27.18	26.99	26.60	26.45	26.35	26.32	26.41	26.28	26.33	26.37
	C4	31.99	33.43	32.00	31.99	31.31	31.42	31.20	31.27	31.30	31.26	31.14	31.31
GPT-2 Medium	WikiText-2	19.77	19.93	19.80	19.79	19.57	19.35	19.31	19.34	19.34	19.32	19.36	19.29
	C4	24.99	25.17	24.92	25.06	25.02	24.85	24.82	24.80	24.86	24.88	24.84	24.75
GPT-2 XL	WikiText-2	15.33	15.27	15.33	15.32	15.30	15.30	15.31	15.31	15.31	15.31	15.31	15.30

Appendix

Table 9: Design-choice ablations: perplexity with exact nonlinear functions.

	Ours (Plonky3)	SP1, fixed point		SP1, FP32
$N$	Prover (s)	Prover (s)	Cycles	Prover (s)	Cycles
64	3.15	25.61	12,794	25.29	107,677
128	3.59	26.00	17,863	27.11	207,150
256	3.28	25.97	28,001	30.50	405,742
512	2.98	26.48	48,268	35.58	806,221
1024	3.58	27.33	88,802	44.61	1,622,618

Appendix

Table 10: Cost of proving the Min-K% decision with and without ZK-friendly quantization (mean of three proofs).

		Llama-3.1-8B		Qwen2.5-7B		Qwen2.5-14B		Qwen3-30B-A3B
Operator / configuration	Mode	Wiki	C4	Wiki	C4	Wiki	C4	Wiki	C4
Full B16	–	1126.8453	380.5457	1496.9087	769.6439	32.4609	18.2997	131.3352	120.6984
RMS InvSqrt	O	1126.7972	380.6613	1528.4988	759.8001	32.6139	18.1795	1.0398	1.0562
	E	0.9993	0.9997	0.9999	0.9978	1.0002	0.9997	1.0034	0.9995
SiLU	O	0.9999	1.0005	0.9998	0.9993	0.9998	1.0003	1.0038	0.9992
	E	1117.7300	382.1309	1415.1683	757.1557	31.7131	18.5148	132.4172	116.0887

Appendix

Table 11: Complete operator-wise LUT ablations. All values are perplexity ratios relative to the corresponding W16A16 LUT-off baseline. O: only the indicated operator family uses a B16 LUT; all other nonlinear functions are exact. E: the full B16 pipeline uses the exact function only for the indicated operator family.

	Proving time (seconds)			Analysis: GeLU $B$ , Exp $B$
$B$	GeLU $B$ Exp 16	GeLU 16 Exp $B$	GeLU $B$ Exp $B$	LUT payload (MiB)	Direct queries (M)	All queries (M)	Joint table proof (s)
8	103.58 (0.77)	104.70 (0.53)	104.75 (0.37)	0.102	0.823	18.238	0.483
12	103.37 (0.34)	104.47 (1.25)	103.28 (0.73)	1.625	0.823	17.906	0.592
16	102.65 (3.04)	102.65 (3.04)	102.65 (3.04)	26.000	0.823	17.104	1.325
24	248.44 (3.51)	120.49 (1.13)	265.54 (2.93)	6,656.000	0.823	18.218	103.761

Appendix

Table 12: LUT cost sensitivity on GPT-2 Small.

Direct measurements
		Proving time (seconds)		Peak RAM usage (GiB)
Projection	Weight shape	Floating-point	Integer	Floating-point	Integer
Q/K/V/O, gate/up	$5120\times 8$	18.998	1.115	14.8816	0.2048
FFN down	$13824\times 8$	42.352	1.359	40.7634	0.2943
Full-width costs: floating-point estimated, integer measured
Q	$5120\times 5120$	12,158.78	11.877	–	–

Appendix

Table 13: Linear proving costs for one Qwen2.5-14B block.

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Post-training quantization (PTQ) has become an important technique for reducing the inference cost of Large Language Models (LLMs). While recent mixed-precision methods improve ultra-low bit quantization by preserving critical subspaces in high precision, they typically construct these subspaces relying solely on activation statistics. This ignores the fundamental nature of linear operations, where the output perturbation is jointly driven by both activation and weight quantization noise. In this paper, we propose CoQuant, a joint weight-activation subspace projection method. By theoretically modeling the expected output error, CoQuant formulates a closed-form weighted PCA solution that balances activation and weight covariances to select the optimal high-precision subspace. Extensive experiments on Llama-3.2 and Qwen2.5 models show that CoQuant consistently outperforms strong PTQ baselines in both WikiText perplexity and zero-shot common-sense reasoning accuracy. These results demonstrate that joint weight-activation subspace modeling provides a principled and effective direction for low-bit LLM quantization. The source code is available at https://github.com/Zachary5895/CoQuant.

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