Analyzing refutations of the well known pebbling formulas we prove some new strong connections between pebble games and algebraic proof system, showing that there is a parallelism between the reversible, black and black-white pebbling games on one side, and the three algebraic proof systems NS, MC and PC on the other side. In particular we prove: \begin{itemize} \item For any DAG $G$ with a single sink, if there is a Monomial Calculus refutation for $Peb(G)$ having simultaneously degree $s$ and size $t$ then there is a black pebbling strategy on $G$ with space $s$ and time $t+s$. Also if there is a black pebbling strategy for $G$ with space $s$ and time $t$ it is possible to extract from it a MC refutation for $Peb(G)$ having simultaneously degree $s$ and size $2t(s-1)$. These results are analogous to those proven in [de Rezende et al. 21] for the case of reversible pebbling and Nullstellensatz. Using them we prove degree separations between NS and MC as well as strong degree-size tradeoffs for MC. \item We show that the variable space needed for the refutation of pebbling formulas in Polynomial Calculus exactly coincides with the black-white pebbling number of the corresponding graph. One direction of this result was known. We present an new elementary proof of it. \item We show that for any unsatisfiable CNF formula $F,$ the variable space in a Resolution refutation of the formula is a lower bound for the monomial space in a PCR refutation for the extended formula $F[\oplus]$. This implies that for any DAG $G$, the monomial space needed in the PCR refutation of an XOR pebbling formulas is lower bounded by the black-white pebbling number of the corresponding graph. This solves affirmatively Open Problem 7.11 from [Buss Nordström 21]. \item The last result also proves a strong separation between degree and monomial space in PCR of size $\Omega(\frac{n}{\log n})$ with the additional property that it is independent of the field characteristic. This question was posed in [Filmus et al. 13]. \end{itemize}

Analyzing refutations of the well known pebbling formulas we prove some new strong connections between pebble games and algebraic proof system, showing that there is a parallelism between the reversible, black and black-white pebbling games on one side, and the three algebraic proof systems NS, MC and PC on the other side. In particular we prove: \begin{itemize} \item For any DAG $G$ with a single sink, if there is a Monomial Calculus refutation for $Peb(G)$ having simultaneously degree $s$ and size $t$ then there is a black pebbling strategy on $G$ with space $s$ and time $t+s$. Also if there is a black pebbling strategy for $G$ with space $s$ and time $t$ it is possible to extract from it a MC refutation for $Peb(G)$ having simultaneously degree $s$ and size $2t(s-1)$. These results are analogous to those proven in [de Rezende et al. 21] for the case of reversible pebbling and Nullstellensatz. Using them we prove degree separations between NS and MC as well as strong degree-size tradeoffs for MC. \item We show that the variable space needed for the refutation of pebbling formulas in Polynomial Calculus exactly coincides with the black-white pebbling number of the corresponding graph. One direction of this result was known. We present an new elementary proof of it. \item We show that for any unsatisfiable CNF formula $F,$ the variable space in a Resolution refutation of the formula is a lower bound for the monomial space in a PCR refutation for the extended formula $F[\oplus]$. This implies that for any DAG $G$, the monomial space needed in the PCR refutation of an XOR pebbling formulas is lower bounded by the black-white pebbling number of the corresponding graph. This solves affirmatively Open Problem 7.11 from [Buss Nordström 21]. \item The last result also proves a strong separation between degree and monomial space in PCR of size $\Omega(\frac{n}{\log n})$ with the additional property that it is independent of the field characteristic. This question was posed in [Filmus et al. 13]. \end{itemize}