I can provide the following results relating to the question about lower central series quotients and group cohomology.

Denote $gr(G;\mathbb{Q}):=\oplus_{i\geq 0}G_i/G_{i+1}\otimes_{\mathbb{Z}} \mathbb{Q}$ the associated graded Lie algebra of $G$.

In some specially cases, there is a close relationship between the lower central series quotients $G_r/G_{r+1}$ (away from torsion) and the group cohomology: $$H^*(G;\mathbb{Q})^!\cong Ext^*_U(\mathbb{Q},\mathbb{Q}).$$ Here, $H^*(G;\mathbb{Q})^!$ is the quadratic dual, and $U$ is the universal enveloping algebra of $gr(G;\mathbb{Q})$. For example, when $G$ is the classical pure braid groups $P_n$, the above isomorphism is valid. Another family of examples is the virtual pure braid groups $vP_n$ and their subgroups $vP_n^+$, which are also known as the quasitriangular group $QTr_n$ and triangular group $Tr_n$. See Paper of Bartholdi- Enriquez- Etingof- Rains, arXiv:math/0509661v6. In order to have the isomorphism, the following two conditions are necessary: the group $G$ is graded-formal(gr(G;\mathbb){Q}) is quadratic) and the cohomology algebra $H^*(G;\mathbb{Q})$ is Koszul.

Partial relation between the lower central series quotients and Massey products can be found in paper of Fenn and Sjerve: ''Massey products and lower central series of free groups''.

Relating question see: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra

I can provide the following results relating to the question about lower central series quotients and group cohomology.

Denote $gr(G;\mathbb{Q}):=\oplus_{i\geq 0}G_i/G_{i+1}\otimes_{\mathbb{Z}} \mathbb{Q}$ the associated graded Lie algebra of $G$.

In some specially cases, there is a close relationship between the lower central series quotients $G_r/G_{r+1}$ (away from torsion) and the group cohomology: $$H^*(G;\mathbb{Q})^!\cong Ext^*_U(\mathbb{Q},\mathbb{Q}).$$ Here, $H^*(G;\mathbb{Q})^!$ is the quadratic dual, and $U$ is the universal enveloping algebra of $gr(G;\mathbb{Q})$. For example, when $G$ is the classical pure braid groups $P_n$, the above isomorphism is valid. Another family of examples is the virtual pure braid groups $vP_n$ and their subgroups $vP_n^+$, which are also known as the quasitriangular group $QTr_n$ and triangular group $Tr_n$. See Paper of Bartholdi- Enriquez- Etingof- Rains, arXiv:math/0509661v6. In order to have the isomorphism, the following two conditions are necessary: the group $G$ is graded-formal(gr(G;\mathbb){Q}) is quadratic) and the cohomology algebra $H^*(G;\mathbb{Q})$ is Koszul.

Partial relation between the lower central series quotients and Massey products can be found in paper of Fenn and Sjerve: ''Massey products and lower central series of free groups''.

Relating question see: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra

I can provide the following results relating to the question about lower central series quotients and group cohomology.

Denote $gr(G;\mathbb{Q}):=\oplus_{i\geq 0}G_i/G_{i+1}\otimes_{\mathbb{Z}} \mathbb{Q}$ the associated graded Lie algebra of $G$.

In some specially cases, there is a close relationship between the lower central series quotients $G_r/G_{r+1}$ (away from torsion) and the group cohomology: $$H^*(G;\mathbb{Q})^!\cong Ext^*_U(\mathbb{Q},\mathbb{Q}).$$ Here, $H^*(G;\mathbb{Q})^!$ is the quadratic dual, and $U$ is the universal enveloping algebra of $gr(G;\mathbb{Q})$. For example, when $G$ is the classical pure braid groups $P_n$, the above isomorphism is valid. Another family of examples is the virtual pure braid groups $vP_n$ and their subgroups $vP_n^+$, which are also known as the quasitriangular group $QTr_n$ and triangular group $Tr_n$. See Paper of Bartholdi- Enriquez- Etingof- Rains, arXiv:math/0509661v6. In order to have the isomorphism, the following two conditions are necessary: the group $G$ is 1-formal and the cohomology algebra $H^*(G;\mathbb{Q})$ is Koszul.

Partial relation between the lower central series quotients and Massey products can be found in paper of Fenn and Sjerve: ''Massey products and lower central series of free groups''.

Relating question see: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra

I can provide the following results relating to the question about lower central series quotients and group cohomology.

Denote $gr(G;\mathbb{Q}):=\oplus_{i\geq 0}G_i/G_{i+1}\otimes_{\mathbb{Z}} \mathbb{Q}$ the associated graded Lie algebra of $G$.

In some specially cases, there is a close relationship between the lower central series quotients $G_r/G_{r+1}$ (away from torsion) and the group cohomology: $$H^*(G;\mathbb{Q})^!\cong Ext^*_U(\mathbb{Q},\mathbb{Q}).$$ Here, $H^*(G;\mathbb{Q})^!$ is the quadratic dual, and $U$ is the universal enveloping algebra of $gr(G;\mathbb{Q})$. For example, when $G$ is the classical pure braid groups $P_n$, the above isomorphism is valid. Another family of examples is the virtual pure braid groups $vP_n$ and their subgroups $vP_n^+$, which are also known as the quasitriangular group $QTr_n$ and triangular group $Tr_n$. See Paper of Bartholdi- Enriquez- Etingof- Rains, arXiv:math/0509661v6. In order to have the isomorphism, the following two conditions are necessary: the group $G$ is graded-formal(gr(G;\mathbb){Q}) is quadratic) and the cohomology algebra $H^*(G;\mathbb{Q})$ is Koszul.

Partial relation between the lower central series quotients and Massey products can be found in paper of Fenn and Sjerve: ''Massey products and lower central series of free groups''.

Relating question see: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra