Proof systems
This post is a reframing of the evolution of proof systems from the contemporary perspective of LDE polynomial commitment protocols.
While proof systems have thus far always been presented endtoend, they often break down into two parts: an arithmetization technique and a polynomial commitment scheme. Despite being presented integral, these are mostly interchangeable and it is perfectly reasonable to for example use Plonk arithmetization on the FRI commitment protocol.
The choice of Polynomial Commitment scheme determines the field $\F$ that is used in the arithmetization layer. While it is possible to use math outside of $\F$ in the claim, this requires costly emulation using $\F$math. Much research has been devoted to making this emulation more efficient.
Basic Architecture
Claims
Arithmetization
Polynomial Commitment Protocols
Commitment Schemes
Interactive proofs
FiatShamir
Noninteractive proofs.
Arithmetization
Pinocchio
This is the system used by Circom, Zorkates, Groth16 and most Ethereum projects that do not develop their own proof system.
The computation is first reduced to an algebraic circuit of basic operations over a field. The circuit is represented by a triplet of matrices $(A, B, C) ∈ \F^{n × m}$ such that a valid witness $\vec s ∈ \F^n$ satisfies.
$n$ is the number of variables in the system, $m$ is the number of gate relationships.
$$ \vec s ⋅ A \circ \vec s ⋅ B  \vec s ⋅ C = 0 $$
We pick a basis $\vec x ∈ \F^m$ and compute $3⋅n + 1$ polynomials $A_i, B_i, C_i, Z ∈ \F[X_{<m}]$ such that
$$ \begin{aligned} A_i(x_j) &= A_{ij} & B_i(x_j) &= B_{ij} & C_i(x_j) &= C_{ij} & Z(x_i) &= 0 \end{aligned} $$
Given a witness $\vec s$, we then compute $A, B, C ∈ \F[X_{<m}]$ such that
$$ \begin{aligned} A(X) &= \sum_i s_i ⋅A_i(X) & B(X) &= \sum_i s_i ⋅B_i(X) & C(X) &= \sum_i s_i ⋅C_i(X) \end{aligned} $$
The circuit is satisfied iff there exists a $H ∈ \F[X_{<m}]$ such that
$$ A(X) ⋅ B(X)  C(X) = H(X) ⋅ Z(X) $$

Bryan Parno, Craig Gentry, Jon Howell, and Mariana Raykova (2013). "Pinocchio: Nearly Practical Verifiable Computation" link

http://www.zeroknowledgeblog.com/index.php/thepinocchioprotocol
Groth16
 Jens Groth (2016). "On the Size of Pairingbased Noninteractive Arguments" link
Sonic
 Mary Maller,d Sean Bowe, Markulf Kohlweiss, and Sarah Meiklejohn (2019). "Sonic: ZeroKnowledge SNARKs from LinearSize Universal and Updateable Structured Reference Strings" link
Stark
 Eli BenSasson, Iddo Bentov, Yinon Horesh, and Michael Riabzev (2018). "Scalable, transparent, and postquantum secure computational integrity" link
Plonk

Ariel Gabizon, Zachary J. Williamson, and Oana Ciobotaru (2019). "PLONK: Permutations over Lagrangebases for Oecumenical Noninteractive arguments of Knowledge" link

Ariel Gabizon, and Zachary J. Williamson (2020). " plookup: A simplified polynomial protocol for lookup tables" link``

Ariel Gabizon and Zachary J. Williamson (2020). "" link

Ariel Gabizon and Zachary J. Williamson (2021). "fflonk: a FastFourier inspired verifier efficient version of PlonK" link
Polynomial Commitment
https://hackernoon.com/kzg10ipafrianddarksanalysisofpolynomialcommitmentschemes
KZG
We want to proof
$$ \forall_i F(x_i) = 0 $$
on some domain $\vec x ∈ \F^n$. We construct the unique $Z(X) ∈ \F[X_{<n}]$ such that $Z(x_i) = 0$.
$$ F(X) = 0 $$
Pick a random $z ∈ \F$.
https://www.cryptologie.net/article/526/malleroptimizationtoreduceproofsize/
 Aniket Kate, Gregory M. Zaverucha, and Ian Goldberg (2010). "ConstantSize Commitments to Polynomials and Their Applications" link
IPA
 Benedikt Bünz, Jonathan Bootle, Dan Boneh, Andrew Poelstra, Pieter Wuille, and Greg Maxwell (2017). "Bulletproofs: Short Proofs for Confidential Transactions and More" link
FRI
 Eli BenSasson, Iddo Bentov†, Yinon Horesh, and Michael Riabzev (2017). "Fast ReedSolomon Interactive Oracle Proofs of Proximity" [link](<https://eccc.weizmann.ac.il/report/2017/134/revision/2/download/)
 Eli BenSasson, Lior Goldberg, Swastik Kopparty, and Shubhangi Saraf (2019). "DEEPFRI: Sampling Outside the Box Improves Soundness" link https://eprint.iacr.org/2020/654
DARK
Implementations
https://minaprotocol.com/blog/kimchithelatestupdatetominasproofsystem
https://eprint.iacr.org/2016/116
https://eprint.iacr.org/2018/828